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Vectors Part Trois

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Vector Geometry Consider this parallelogram Q O P R a b Opposite sides are Parallel OQ is known as the resultant of a and b

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Resultant of Two Vectors Is the same, no matter which route is followed Use this to find vectors in geometrical figures

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Example Q O P R a b. S S is the Midpoint of PQ. Work out the vector = a + ½b

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Alternatively Q O P R a b. S S is the Midpoint of PQ. Work out the vector = a + ½b = b + a - ½b = ½b + a

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Example A B C p q M M is the Midpoint of BC Find BC AC= p, AB = q BCBAAC=+ = -q + p = p - q

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Example A B C p q M M is the Midpoint of BC Find BM AC= p, AB = q BM ½BC = = ½(p – q)

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Example A B C p q M M is the Midpoint of BC Find AM AC= p, AB = q = q + ½(p – q) AM + ½BC = AB = q +½p - ½q = ½q +½p= ½(q + p)= ½(p + q)

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Alternatively A B C p q M M is the Midpoint of BC Find AM AC= p, AB = q = p + ½(q – p) AM + ½CB = AC = p +½q - ½p = ½p +½q= ½(p + q)

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Why?? We represent objects using mainly linear primitives: points points lines, segments lines, segments planes, polygons planes, polygons Need to know how to compute distances, transformations…

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Basic definitions Points specify location in space (or in the plane). Points specify location in space (or in the plane). Vectors have magnitude and direction (like velocity). Vectors have magnitude and direction (like velocity). Points Vectors

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Point + vector = point

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vector + vector = vector Parallelogram rule

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point - point = vector A B B – A A B A – B Final - Initial

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point + point: not defined!! Makes no sense

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Map points to vectors If we have a coordinate system with origin at point O We can define correspondence between points and vectors:

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Position Vectors A position vector for a point X (with respect to the origin 0) is the fixed vector O X

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Components of a Vector Between Two Points Given and, then Proof:

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Understanding

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Understanding

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Understanding

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Understanding Determine the coordinates of the point that divides AB in the ratio 5:3 where A is (2, -1, 4) and B is (3, 1, 7) If AP:PB = m:n then

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Understanding Consider collinear points P, Q, and R. consider also a reference point O. Write OQ as a linear combination of OP and OR if: a)Q divides PR (internally) in the ratio 3:5 b)Q divides PR (externally) in the ratio -2:7 PQ R 0 35 Since: Then:

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Understanding Consider collinear points P, Q, and R. consider also a reference point O. Write OQ as a linear combination of OP and OR if: a)Q divides PR (internally) in the ratio 3:5 b)Q divides PR (externally) in the ratio -2:7 QP R 0 -25 Since: Then: 7

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Understanding Determine the point Q that divides line segment PR in the ratio 5:8 if P is (1,5,2) and R is (-4,1,2).

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Inner (dot) product Defined for vectors: Defined for vectors: L v w w·cos(θ) is the scalar projection of w on vector v, which we will call L the dot product can be understood geometrically as the product of the length of this projection and the length of v. The dot product or scalar product allows us to determine a product the magnitude of one vector by the magnitude of the component of the other vector that’s parallel to the first. (The dot product was invented specifically for this purpose.)

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Dot product in coordinates v w xvxv yvyv xwxw ywyw x y O An interesting proof

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Proof of Geometric and Coordinate Definitions Equality From the Cosine Law therefore

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Vector Properties

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Understanding

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Understanding Prove that for any vector, Proof:

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Understanding Cosine has the range of -1 to 1

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Understanding Given that work is defined as (the dot product of force, f, and distance travelled, s. A crate on a ramp is hauled 8 m up the ramp under a constant force of 20 N applied at an angle of 30 0 to the ramp. Determine the work done.

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Understanding The angle between two vectors u and v is 110 0, if the magnitude of vector u is 12, then determine: The projection of vector u onto vector v

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The Cross Product Some physical concepts (torque, angular momentum, magnetic force) require that we multiply the magnitude of one vector by the magnitude of the component of the other vector that is perpendicular to the first (recall in the dot product we used the parallel component). The cross product was invented for this specific purpose B Bsin(θ) θ A The magnitude (length) of the cross product

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The Cross Product Unlike the dot product, the cross product is a vector quantity. Where θ is the angle between and, and is a unit vector perpendicular to both and such that and form a right handed triangle. Take your right hand, and point all 4 fingers (not the thumb) in the direction of the first vector x. Next, rotate your arm or wrist so that you can curl your fingers in the direction of y. Then extend your thumb, which is perpendicular to both x and y. The thumb’s direction is the direction the cross product will point.

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Understanding Into board Out off board

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Understanding First suppose u and v are collinear, then the angle, θ, between u and v is 0 0. Since sin(0 0 )=0, then: Conversely suppose that then. Since u and v are non-zero vectors, then sin(θ)=0, and hence θ=0 or θ= π. Therefore u and v are collinear.

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Properties

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Understanding Determine the area of the parallelogram below, using the cross product. 8 13 30 0

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Understanding State whether each expression has meaning. If not, explain why, if so, state whether it is a vector or a scalar quantity. scalar Non sense vector scalar

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Understanding A right-threaded bolt is tightened by applying a 50 N force to a 0.20 m wrench as shown in the diagram. Find the moment of the force about the centre of the bolt. The moment of the bolt is 9.4 Nm and is directed down

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Vector Product in Component Form Recall: But how do we obtain the answer in component form? The easiest technique is called the Sarrus’ Scheme

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Understanding

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Understanding A D B C a.

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Understanding b. Area of triangle is one-half the area of the parallelogram

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Understanding

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Understanding Find a unit vector that is perpendicular to both: We need to apply the cross product to determine a vector perpendicular to both u and v. Then we divide by the length of this vector to make it a unit vector

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Useful Properties

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