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14.1Vectors in Three-dimensional Rectangular Coordinate System 14.2Vector Product and Scalar Triple Product Chapter Summary Case Study Vectors in Three-dimensional Space 1414

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P. 2 A plane is approaching Hong Kong International Airport. The flight- control operator of the control tower is trying to give instructions to the pilot to provide a safe route for landing. Case Study To describe the position and the route of the plane, we can introduce the three-dimensional rectangular coordinate system as shown in the figure. Let the airport be the origin of the coordinate system. Then we use the triplet (x, y, z) to describe the horizontal position (x, y as in the two-dimensional case) and the height of the plane (z). The plane is located at the point A(2, 1, 4), B(1, 2, 3), C(2, 0, 2), D(1, 1, 1) are points in space, such that the plane follows the route A B C D O. We are now ready for landing. Please indicate a flight route. Captain, please follow the following flight route.

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P Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System A. Vectors in Three-dimensional Space The addition, subtraction, scalar multiplication, negative, parallelism of vectors, and the rules of operations of vectors are also defined in the same way as in the case of plane vectors. For any two points A and B in space, the directed line segment from A to B is called the vector from A to B, and is denoted by. The magnitude of is denoted by, which is the same as the vectors on the plane defined before. For example, for the cube ABCDEFGH, we have 1.(equal vectors) 2.(negative vectors) 3.(parallel vectors) 4.(addition of vectors ) 5.(subtraction of vectors)

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P. 4 Example 14.1T Solution: (a) (b) (c) 14.1 Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System A. Vectors in Three-dimensional Space The figure shows a cube ABCDEFGH. Let = a, = b and = c. Express the following in terms of a, b and c. (a)(b)(c)

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P. 5 Example 14.2T Solution: 14.1 Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System A. Vectors in Three-dimensional Space The figure shows a cube ABCDEFGH. Prove that

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P Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System B. Representation of Vectors in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System The directions of these axes are aligned in such a way that they obey the right-hand rule. The three-dimensional Cartesian coordinate system R 3 consists of three mutually perpendicular axes: x, y and z. If the x- and y-axes are represented by the index finger and the middle finger respectively, then the thumb represents the z-axis.

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P Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System B. Representation of Vectors in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System These three values are called x-, y- and z-coordinates of the point respectively. Every point in space can be represented in the three- dimensional coordinate system by the triplet (x, y, z), where x, y and z represent the directed distances from the yz-, zx- and xy-planes respectively. The point of intersection of the three axes is called the origin O and its coordinates are (0, 0, 0). In the figure, i, j and k are the unit vectors in the positive directions of x-, y- and z-axes respectively. They have a common starting point at the origin, and their terminal points are (1, 0, 0), (0, 1, 0) and (0, 0, 1) respectively. P(x, y, z) is a point in R 3, so we can express the position vector as By Pythagoras theorem, we have

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P Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System B. Representation of Vectors in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System Now if two points A(x 1, y 1, z 1 ) and B(x 2, y 2, z 2 ) in R 3 are given, the vector from A to B can be found by subtracting the position vector from which is the same as we did in the case of R 2, then we have

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P. 9 Example 14.3T Solution: 14.1 Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System B. Representation of Vectors in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System Given two points P(–3, –2, 8) and Q(0, –5, 4). Find the unit vector in the direction of. Unit vector

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P Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System B. Representation of Vectors in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System Property 14.1 (a) pi + qj + rk = si + tj + uk if and only if p = s, q = t and r = u, and (b) pi + qj + rk = 0 if and only if p = q = r = 0. Consider two vectors pi + qj + rk and si + tj + uk in R 3. As i, j and k are non parallel vectors, we have

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P. 11 Example 14.4T Solution: Given three points A(–3, 1, 5), B(2, 5, –1) and C(–6, 4, 3). Find the coordinates of a point D if (a)ABCD forms a parallelogram, (b)ABDC forms a parallelogram Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System B. Representation of Vectors in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System Let the coordinates of D be (x, y, z). (a) If ABCD forms a parallelogram, We have The coordinates of D are ( 11, 0, 9).

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P. 12 Example 14.4T Solution: Given three points A(–3, 1, 5), B(2, 5, –1) and C(–6, 4, 3). Find the coordinates of a point D if (a)ABCD forms a parallelogram. (b)ABDC forms a parallelogram Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System B. Representation of Vectors in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System (b) If ABDC forms a parallelogram, We have The coordinates of D are ( 1, 8, 3).

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P. 13 Example 14.5T Solution: Consider the three vectors a = 3i – 4j + 2k, b = i – 3j – k and c = 5i + 2j + k. If m = –16i – 8j – 7k, express m in terms of a, b and c Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System B. Representation of Vectors in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System Let m = a + b + c. Consider the determinant of the coefficient matrix: By Cramers rule,

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P. 14 Example 14.6T Solution: 14.1 Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System B. Representation of Vectors in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System (a) (b)(b) Given two points A and B with = 3i – 8j + 5k and = 6i + 2j – 7k. C is a point on the line segment AB. Find if (a)C is the mid-point of AB, (b)C divides AB in the ratio 2 : 1.

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P Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System C. Scalar Product In the three-dimensional rectangular coordinate system R 3, as the unit base vectors i, j and k are mutually perpendicular, we have i i = j j = k k = 1 i j = j i = 0 j k = k j = 0 i k = k i = 0 If a = x 1 i + y 1 j + z 1 k and b = x 2 i + y 2 j + z 2 k are two non-zero vectors, thena b = x 1 x 2 + y 1 y 2 + z 1 z 2, where is the angle between a and b. We also have the following properties of scalar product: a b = b a a (b + c) = a b + a c (ka) b = k(a b) = a (kb)

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P. 16 Example 14.7T Solution: Two vectors r = 2i + 3j – k and s = i + 2k are given. (a)Find the value of r s. (b)Hence find the angle between r and s Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System C. Scalar Product (a) (b)(b) The angle between r and s is 90.

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P. 17 Example 14.8T Solution: If the vectors ci + 5j – 3k and 2ci + cj + k are perpendicular to each other, find the value(s) of c. ci + 5j – 3k and 2ci + cj + k are perpendicular to each other Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System C. Scalar Product

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P. 18 Example 14.9T Solution: Given three points A(2, 1, 6), B(– 5, 3, 5) and C(0, –6, 5) are vertices of ABC. Solve ABC. (Give the answers in surd form or correct to the nearest degree.) 14.1 Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System C. Scalar Product (cor. to the nearest degree)

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P. 19 Example 14.9T Solution: Given three points A(2, 1, 6), B(– 5, 3, 5) and C(0, –6, 5) are vertices of ABC. Solve ABC. (Give the answers in surd form or correct to the nearest degree.) 14.1 Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System C. Scalar Product (cor. to the nearest degree)

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P. 20 Example 14.10T 14.1 Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System C. Scalar Product Solution: Given two vectors x = 6j – 5k and y = 3i – 4j. (a)Find the angle between x and y, correct to the nearest degree. (b)Find the length of the projection of y on x. Let be the angle between x and y. (a) (cor. to the nearest degree) The angle between x and y is 128. (b) The length of the projection of y on x

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P. 21 Suppose we have two non-zero vectors a and b in the three-dimensional space. The vector product of a and b, denoted by a b, is the vector which is perpendicular to both a and b, with the magnitude equal to |a b| = |a||b|sin, where is the angle between a and b (with 0° 180°) Vector Product and Scalar Triple Product Triple Product A. Definition of Vector Product Note: 1.a b is read as a cross b. Therefore the vector product is also called the cross product. 2. The vector product is only defined in the three-dimensional space. 3. In contrast to the scalar product of two vectors, the vector product is a vector while the scalar product is a scalar. a b = |a||b|sin, where is the angle between a and b, and is a unit vector whose direction is defined by the right-hand rule. Particularly, the direction of a b is defined in such a way that a, b and a b always obey the right-hand rule. In conclusion,

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P Vector Product and Scalar Triple Product Triple Product A. Definition of Vector Product In particular, if b = a, we have For the unit vectors i, j and k: For and two non-zero vectors a and b, a b = 0 if and only if a and b are parallel to each other. a a = 0. i i = j j = k k = 0 i j = kj i = k j k = ik j = i k i = ji k = j

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P Vector Product and Scalar Triple Product Triple Product B. Properties of Vector Product Proof of (a): b a = (a b) Property 14.2Properties of Vector Product (a)b a = (a b) (b)(a + b) c = a c + b c (c)a (b + c) = a b + a c (d)(ka) b = a (kb) = k(a b) (e)|a b| 2 = |a| 2 |b| 2 – (a b) 2

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P Vector Product and Scalar Triple Product Triple Product B. Properties of Vector Product Proof of (d): If k = 0 or a = 0 or b = 0, then (ka) × b = a × (kb) = (k a × b) = 0. Assume that k 0 and a and b are non-zero. When k < 0, When k > 0, Similarly, it can be proved that a × (kb) = k(a × b). (ka) × b = a × (kb) = k(a × b) Let be the angle between a and b, and be the unit vector in the direction of a × b.

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P Vector Product and Scalar Triple Product Triple Product B. Properties of Vector Product Proof of (e): |a × b| 2 = (|a||b|sin ) 2 Since |a × b| = |a||b|sin = |a| 2 |b| 2 sin 2 = |a| 2 |b| 2 |a| 2 |b| 2 cos 2 = |a| 2 |b| 2 – (a b) 2 sin 2 = 1 – cos 2 a – b = |a||b|cos

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P Vector Product and Scalar Triple Product Triple Product C. Calculation of Vector Product We can use the determinant to represent the vector product: If a = x 1 i + y 1 j + z 1 k and b = x 2 i + y 2 j + z 2 k, then

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P. 27 Example 14.11T Solution: For the following pairs of vectors m and n, find the vector products m × n. (a)m = 3i + 8j, n = 6k (b)m = –4i + 2j + 6k, (a) (b) 14.2 Vector Product and Scalar Triple Product Triple Product C. Calculation of Vector Product

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P. 28 Example 14.12T Solution: 14.2 Vector Product and Scalar Triple Product Triple Product C. Calculation of Vector Product P, Q and R are three points with position vectors i + j + k, –2j and –i + 3j – k respectively. Find the unit vectors which are perpendicular to and. Unit vectors which are perpendicular to and

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P Vector Product and Scalar Triple Product Triple Product D. Applications of Vector Product Consider a parallelogram ABCD. Area of the parallelogram ABCD = Since the area of ABD is half that of parallelogram ABCD, we can obtain a formula for the area of triangle: Area of ABD = The above formula can be further rewritten as Area of ABD =

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P. 30 Example 14.13T Solution: Find the area of the triangle formed by vertices X(2, 1, 1), Y(0, –1, 0) and Z(–2, 1, –1) Vector Product and Scalar Triple Product Triple Product D. Applications of Vector Product Area of XYZ

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P. 31 Since the scalar product of two vectors is a scalar, thus a (b c) is a scalar as the name suggests. The volume of a parallelepiped (a prism with all faces are parallelograms) with sides a, b and c is given by |a (b × c)| Vector Product and Scalar Triple Product Triple Product E. Scalar Triple Product The expression a (b c) is called the scalar triple product of a, b and c. In the three-dimensional rectangular coordinate system, suppose a = x 1 i + y 1 j + z 1 k, b = x 2 i + y 2 j + z 2 k and c = x 3 i + y 3 j + z 3 k, then Note: If a (b c) = 0, the volume of the parallelepiped with sides a, b and c equals zero. This only when a, b and c are coplanar.

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P Vector Product and Scalar Triple Product Triple Product E. Scalar Triple Product Property 14.3Properties of Scalar Triple Product (a)(a b) c = a (b c) (b)a (b c) = b (c a) = c (a b) Since a determinant is unchanged when interchanging the rows twice and for any non-zero vectors x and y, x y = y x, we have the following properties of the scalar triple product:

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P. 33 Example 14.14T Solution: If p = 2i + j + 3k, q = 3i – j – 2k and r = –i + 2j – k, find (a)r × p, and (b)q (r × p). (a) (b)(b) 14.2 Vector Product and Scalar Triple Product Triple Product E. Scalar Triple Product

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P. 34 Example 14.15T Solution: 14.2 Vector Product and Scalar Triple Product Triple Product E. Scalar Triple Product Consider A(2, 1, 0), B(–3, 4, 5), C(0, –2, 4) and D(1, 2, 5). Find the volume of the parallelepiped with sides, and. Volume of the parallelepiped

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P Vectors in Three-dimensional Rectangular Coordinate System Chapter Summary 1.Every point in the space can be represented in the three-dimensional coordinate system by the triplet (x, y, z), where x, y and z represent the directed distances from the yz-, zx- and xy-planes respectively. 2.The distance between two points A(x 1, y 1, z 1 ) and B(x 2, y 2, z 2 ) is given by

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P. 36 Chapter Summary 14.1 Vectors in Three-dimensional Rectangular Coordinate System 1.The rules of operations and properties of vectors in the space are the same as vectors on a plane. 2.In R 3, we define three mutually perpendicular unit vectors i, j and k, which point in the positive direction of x-, y- and z-axes respectively. 3.For a point P(x, y, z) in R 3, the position vector can be expressed as, where.

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P. 37 Chapter Summary 14.1 Vectors in Three-dimensional Rectangular Coordinate System Scalar Product If a = x 1 i + y 1 j + z 1 k and b = x 2 i + y 2 j + z 2 k, are two non-zero vectors, then where is the angle between a and b.

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P Vector Product and Scalar Triple Product Chapter Summary Vector Product 1.If a = x 1 i + y 1 j + z 1 k and b = x 2 i + y 2 j + z 2 k, are non-zero vectors and is the angle between them, then 2. Area of ABC

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P. 39 Scalar Triple Product 14.2 Vector Product and Scalar Triple Product Chapter Summary 2. Volume of the parallelepiped with sides a, b and c = |a (b c)|. 1.If a = x 1 i + y 1 j + z 1 k and b = x 2 i + y 2 j + z 2 k and c = x 3 i + y 3 j + z 3 k are non-zero vectors, then

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Follow-up 14.1 Solution: (a) (b) (c) 14.1 Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System A. Vectors in Three-dimensional Space The figure shows a cube ABCDEFGH. Let = a, = b and = c. Express the following in terms of a, b and c. (a)(b)(c)

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Follow-up 14.2 The figure shows a tetrahedron ABCD. M and N are the mid-points of BC and AD respectively. Prove that Solution: 14.1 Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System A. Vectors in Three-dimensional Space

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Follow-up 14.3 Solution: 14.1 Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System B. Representation of Vectors in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System Given two points M(–1, 4, 2) and N(3, 1, –3). Find the unit vector in the direction of. Unit vector

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Solution: Follow-up 14.4 Given three points A(1, 4, 2), B(6, 3, 5) and C(3, 6, 3). Find the coordinates of a point D if ABCD forms a parallelogram Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System B. Representation of Vectors in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System Let the coordinates of D be (x, y, z). If ABCD forms a parallelogram, We have The coordinates of D are ( 8, 7, 4).

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Follow-up 14.5 Solution: Given three vectors a = – 2i + 4j + k, b = i – 2j + 5k and c = 4i + j – 3k. If r = 14i + 8j, express r in terms of a, b and c Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System Let r = a + b + c. Consider the determinant of the coefficient matrix: By Cramers rule, B. Representation of Vectors in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System

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Solution: Follow-up 14.6 Given two points A and B with = –5i + 2j – 4k and = 3i + 8j + 8k. C is a point on the line segment AB. Find if (a)C is the mid-point of AB, (b)C divides AB in the ratio 3 : 2. (a) (b)(b) 14.1 Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System B. Representation of Vectors in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System

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Follow-up 14.7 Solution: Given two vectors m = i – 2j + 4k and n = 3i + 4j + 3k. (a) Find the value of m n. (b) Hence find the angle between m and n, correct to the nearest degree Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System C. Scalar Product (a) (b) Let be the angle between m and n. (cor. to the nearest degree) The angle between m and n is 75.

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Follow-up 14.8 Solution: If the vectors bi – 3j + b 2 k and i + 2bj – 3k are perpendicular to each other, find the value(s) of b Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System C. Scalar Product and are perpendicular to each other.

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Follow-up 14.9 Solution: Three points A(3, 3, 4), B(0, 5, 2) and C(4, 1, 7) are vertices of ABC. Solve ABC. (Give the answers in surd form or correct to the nearest degree.) 14.1 Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System C. Scalar Product

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Follow-up 14.9 Solution: 14.1 Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System C. Scalar Product (cor. to the nearest degree) Three points A(3, 3, 4), B(0, 5, 2) and C(4, 1, 7) are vertices of ABC. Solve ABC. (Give the answers in surd form or correct to the nearest degree.)

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Follow-up Solution: Given two vectors a = i + 3j + 5k and b = 2i + 4j + 6k. (a)Find the angle between a and b, correct to the nearest degree. (b)Find the length of the projection of a on b Vector in Three-dimensional Rectangular Coordinate System Rectangular Coordinate System C. Scalar Product Let be the angle between a and b. (cor. to the nearest degree) (b) The length of the projection of a on b (a)

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Follow-up Solution: For the following pairs of vectors p and q, find the vector products p × q. (a)p = 3i + j + 2k, q = i + 4j – k (b)p = i – 4j – 6k, q = 2i + 5k (c)p = 2i – 5j + k, q = –6i + 15j – 3k 14.2 Vector Product and Scalar Triple Product Triple Product C. Calculation of Vector Product (a) (b)(b) (c)(c)

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Follow-up Solution: 14.2 Vector Product and Scalar Triple Product Triple Product C. Calculation of Vector Product Unit vectors which are perpendicular to and X, Y and Z are three points with position vectors i – k, 3i + j – 2k and – j + k respectively. Find the unit vectors which are perpendicular to and.

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Follow-up Solution: Find the area of the triangle with vertices P(0, 1, –1), Q(–2, –1, 1) and R(1, –2, 0) Vector Product and Scalar Triple Product Triple Product D. Applications of Vector Product Area of PQR

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Follow-up Solution: If x = 3i + j + k, y = –i – j + 3k and z = 2i + 2j – 2k, find (a)x × y, and (b)(x × y) z. (a) (b)(b) 14.2 Vector Product and Scalar Triple Product Triple Product E. Scalar Triple Product

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Follow-up Solution: 14.2 Vector Product and Scalar Triple Product Triple Product E. Scalar Triple Product The coordinates of P, Q, R and S are (0, 0, 2), (3, 0, 4), (1, 2, 3) and (0, 1, –2) respectively. Find the volume of the parallelepiped with sides, and. Volume of the parallelepiped

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