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© Boardworks Ltd of 43 © Boardworks Ltd of 43 A2-Level Maths: Core 4 for Edexcel C4.7 Vectors 1 This icon indicates the slide contains activities created in Flash. These activities are not editable. For more detailed instructions, see the Getting Started presentation.

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© Boardworks Ltd of 43 Vectors in two and three dimensions The magnitude of a vector Multiplying vectors by scalars Adding and subtracting vectors Position vectors and coordinate geometry Contents © Boardworks Ltd of 43 Vectors in two and three dimensions

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© Boardworks Ltd of 43 Vectors and scalars A vector is a quantity that has both size (or magnitude) and direction. Examples of vector quantities include: A scalar is a quantity that has size (or magnitude) only. Examples of scalar quantities include: velocity displacement force speed length mass

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© Boardworks Ltd of 43 Representing vectors A vector can be represented using a line segment with an arrow on it. For example, the vector that goes from the point A to the point B can be represented by the following directed line segment. A B The magnitude of the vector is given by the length of the line. The direction of the vector is given by the arrow on the line.

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© Boardworks Ltd of 43 Representing vectors Vectors can also be written using single letters in bold type. For example, we can call this vector a. When this is hand-written, the a is written as a We can write this vector as. To go from the point A to the point B we must move 6 units to the right and 3 units up. A B 6 3

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© Boardworks Ltd of 43 Representing vectors We can also represent vectors in three dimensions relative to a three dimensional coordinate grid: We can represent this movement using a column vector. A third axis, the z -axis, is added at right angles to the xy -plane. Conventionally, we show the z - axis pointing vertically upwards with the xy -plane horizontal. This component tells us the number of units moved in the y -direction. This component tells us the number of units moved in the x -direction.

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© Boardworks Ltd of 43 Representing vectors x y z 5 –3 –2 C D To go from the point C to the point D we must move For example, consider the following three-dimensional vector 5 units in the x -direction, –3 units in the y -direction –2 units in the z -direction. This three-dimensional vector can be written in column vector form as: This component tells us the number of units moved in the y -direction. This component tells us the number of units moved in the x -direction. This component tells us the number of units moved in the z -direction.

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© Boardworks Ltd of 43 Equal vectors Two vectors are equal if they have the same magnitude and direction. For example, in the following diagram: General displacement vectors that are not fixed to any point are often called free vectors. and A B C D

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© Boardworks Ltd of 43 The negative of a vector a A B Suppose the arrow went in the opposite direction, from B to A : We can describe this new vector as: –a–aor Here is the vector This is the negative (or inverse) of the vector A B

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© Boardworks Ltd of 43 The negative of a vector And in three-dimensions, If then In general, If then

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© Boardworks Ltd of 43 The zero and unit vector A vector with a magnitude of 0 is called the zero vector. The zero vector is written as 0 or hand-written as 0 A vector with a magnitude of 1 is called a unit vector. The most important unit vectors are those that run parallel to the x - and y -axes. These are called unit base vectors. The horizontal unit base vector,, is called i.The vertical unit base vector,, is called j.

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© Boardworks Ltd of 43 The unit base vectors The unit base vectors, i and j, run parallel to the x - and y -axes. Any column vector can easily be written in terms of i and j. The number of i’s tells us how many units are moved horizontally, and the number of j’s tells us how many units are moved vertically. For example, y -axis x -axis j i

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© Boardworks Ltd of 43 The unit base vectors In three dimensions, we introduce a third unit base vector, k, that runs parallel to the z -axis. For example, the three-dimensional vector can be written in terms of i, j and k as i is, j is and k is. Vectors written in terms of the unit base vectors i, j and k are usually said to be written in component form. –i + 6j –3k z -axis y -axis x -axis i j k

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© Boardworks Ltd of 43 Vectors in two and three dimensions The magnitude of a vector Multiplying vectors by scalars Adding and subtracting vectors Position vectors and coordinate geometry Contents © Boardworks Ltd of 43 The magnitude of a vector

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© Boardworks Ltd of 43 Finding the magnitude of a vector The magnitude (or modulus) of a vector is given by the length of the line segment representing it. We can calculate this using Pythagoras’s Theorem. For example, suppose we have the vector a A B The magnitude of this vector is written as or a.

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© Boardworks Ltd of 43 Finding the magnitude of a vector

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© Boardworks Ltd of 43 Finding the magnitude of a vector The magnitude of a three-dimensional vector can be found by applying Pythagoras’s Theorem in three dimensions. The magnitude of this vector is given by For example, suppose we have the vector

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© Boardworks Ltd of 43 Finding the magnitude of a vector

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© Boardworks Ltd of 43 The distance between two points If we are given the coordinates of two points and we are asked to find the distance between them we use Pythagoras’ Theorem in the same way. For example, Find the distance between the points with coordinates P (–4, 7, –2) and Q (5, 9, –8). If d is the distance between the points then, using Pythagoras’ Theorem in three dimensions gives: d 2 = (–4 – 5) 2 + (7 – 9) 2 + (–2 – –8) 2 d2 = d 2 = 121 d = 11 In general, if d is the distance between the points ( x 1, y 1, z 1 ) and ( x 2, y 2, z 2 ) then d 2 = ( x 1 – x 2 ) 2 + ( y 1 – y 2 ) 2 + ( z 1 – z 2 ) 2

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© Boardworks Ltd of 43 Unit vectors Remember, if the magnitude of a vector is 1 it is called a unit vector. It is possible to find a unit vector parallel to any given vector, a, by dividing the vector by its magnitude. The unit vector parallel to the vector a is denoted by So, in general, Find a unit vector parallel to b = 4i – j + k

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© Boardworks Ltd of 43 Vectors in two and three dimensions The magnitude of a vector Multiplying vectors by scalars Adding and subtracting vectors Position vectors and coordinate geometry Contents © Boardworks Ltd of 43 Multiplying vectors by scalars

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© Boardworks Ltd of 43 Multiplying vectors by scalars Remember, a scalar quantity can be represented by a single number. It has size but not direction. A vector can be multiplied by a scalar. For example, suppose the vector a is represented as follows: The vector 2a has the same direction but is twice as long. a 2a2a

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© Boardworks Ltd of 43 Multiplying vectors by scalars In general, if the vector is multiplied by the scalar k, then: When a vector is multiplied by a scalar the resulting vector lies either parallel to the original vector or on the same line. For example,

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© Boardworks Ltd of 43 Multiplying vectors by scalars

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© Boardworks Ltd of 43 Vectors in two and three dimensions The magnitude of a vector Multiplying vectors by scalars Adding and subtracting vectors Position vectors and coordinate geometry Contents © Boardworks Ltd of 43 Adding and subtracting vectors

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© Boardworks Ltd of 43 Adding vectors Adding two vectors is equivalent to applying one vector followed by the other. We can represent the addition of these two vectors in the following diagram: a b a + b For example, suppose and.

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© Boardworks Ltd of 43 Adding vectors

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© Boardworks Ltd of 43 Adding vectors When two or more vectors are added together the result is called the resultant vector. In general, if and, then. Given that a = 2i + 6j – k and b = –i + 2j + 7k, find a + b. a + b = (2 –1)i + (6 + 2)j + (–1 + 7)k = i + 8j + 6k

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© Boardworks Ltd of 43 Subtracting vectors We can think of the subtraction of two vectors, a – b, as a + (–b). a b a – b –b a For example, suppose and.

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© Boardworks Ltd of 43 Adding and subtracting vectors

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© Boardworks Ltd of 43 The parallelogram law for adding vectors

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© Boardworks Ltd of 43 Vector arithmetic We have seen that vectors can be multiplied by scalars, added and subtracted. We have also seen that vector addition is commutative. Given that a = 2i – 4j + k and b = j + 2k find 3a – 2b. We can use this to add and subtract any given multiple of a vector given in component or column vector form. For example, 3a – 2b = 3(2i – 4j + k ) – 2(j + 2k ) = 6i – 12j + 3k – 2j – 4k = 6i – 14j – k

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© Boardworks Ltd of 43 Vector arithmetic Start by rearranging the equation to make c the subject. 2c + a = b 2c = b – a Find vector c such that 2c + a = b. Suppose that and. c = (b – a)

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© Boardworks Ltd of 43 A grid of congruent parallelograms

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© Boardworks Ltd of 43 Using vectors to solve problems We can use vectors to solve many problems involving physical quantities such as force and velocity. We can also use vectors to prove geometric results. For example, suppose we have a triangle ABC as follows: A B C The line PQ is such that P is the mid-point of AB and Q is the mid-point of AC. Use vectors to show that PQ is parallel to BC and that the length of BC is double the length of PQ. P Q

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© Boardworks Ltd of 43 Using vectors to solve problems A B C P Q a b We can conclude from this that PQ is parallel to BC and that the length of BC is double the length of PQ. Let’s call vector a and vector b. Therefore,

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© Boardworks Ltd of 43 Collinear points Three or more points are said to be collinear if they lie on the same line. For example Prove that the three points A (–3, 2, 6), B (1, 4, –2) and C (1, 5, –6) are collinear. Since AB is a scalar multiple of BC the two lines must be parallel. They also have the point B in common and so the points A, B and C must be collinear.

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© Boardworks Ltd of 43 Contents © Boardworks Ltd of 43 Position vectors and coordinate geometry Vectors in two and three dimensions The magnitude of a vector Multiplying vectors by scalars Adding and subtracting vectors Position vectors and coordinate geometry

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© Boardworks Ltd of 43 Position vectors A position vector is a vector that is fixed relative to a fixed origin O. For example, suppose the point P has coordinates (4, 6). The position vector of the point P is given by Now, suppose the point Q has coordinates (3, –2). O P q The position vector of the point Q is given by Q Write the vector PQ as a column vector. p

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© Boardworks Ltd of 43 Position vectors O P p q Q To get from the point P to the point Q, we have to go from P to O … … and then from O to Q. P –p So q – p We can check this using the vector diagram. O q Q

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© Boardworks Ltd of 43 Position vectors In general, if A is the point with coordinates ( x 1, y 1 ) and B is the point with coordinate ( x 2, y 2 ) we can write the position vectors and The vector is given by The vector can also be written in terms of i and j as

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© Boardworks Ltd of 43 The mid-point of a line Returning to our example using the points P (4, 6) and Q (3, –2): O P p q Q Let M be the mid-point of the line PQ. M What is the position vector of the point M ?

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© Boardworks Ltd of 43 The mid-point of a line In general, if points A and B have position vectors a and b, then the position vector of the mid-point of the line AB is given by: and so O P p q Q M m

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