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4sinCan you differentiate ?

Aims:• To know the Scalar Product formula and be able to use it to determine if two vector are perpendicular or parallel.• To be able to use the scalar product to find the angle between vectors and direction vectors when a line is given in vector form.

Vectors Lesson 5

The scalar product

The result of this multiplication is a s_________/number quantity, hence the name.

The scalar product of two vectors a and b is defined as:

where θ is the angle between a and b when they are placed tail to tail.

Note that θ is always taken to be between 0° and 180°.

Two vectors can be multiplied together to give the scalar product, also known as the dot product.

= cosa.b a b

In general, if a = a1i + a2j + a3k and b = b1i + b2j + b3k then

a.b = a1b1+ a2b2 + a3b3.

Perpendicular and parallel vectors

If a.b = 0 then a and b are perpendicular, as long as neither are the zero vector.

In particular, for the unit base vectors i, j and k:

We have seen that, since cos 90° = ______:

If two vectors a and b are parallel, a.b = |a||b|.

i.i = j.j = k.k = 1×1 = 1

Also, if two vectors are parallel the angle between them is taken as 0° .

Since cos 0° = ______ we can conclude from the scalar product that:

In particular, for the unit base vectors i, j and k:

i.j = i.k = j.i = j.k = k.i = k.j =

= cosa.b a b

Scalar products of vectors in component form

a.b =

Find the scalar product of

10

a.b =

Since a and b are both non-zero vectors, and their scalar product is _____, they must be p______________________.

Prove that the vectors a = –3i + j + 2k and b = 2i + 8j – k are perpendicular.

1 12 and 41 3

a b

On w/b

a.b =

Find the scalar product of

10

a.b =

Find the scalar product of a = 4i +3j + 2k and b = 4i - 2j – 5k and state whether or not they are perpendicular.

4 01 and 2

12 3a b

Finding the angle between two vectors

To do this we can write the scalar product in the form:

For example,

A useful application of the scalar product is in finding the acute angle between two vectors.

cos a.b=a b

5= 1

2

a

2= 0

3b

On w/b

cos a.b=a b

7= 2

1a

0= 1

2b

1. The position vectors of three points A, B and C relative to an origin O are given respectively by

= 7i + 3j – 3k,

= 4i + 2j – 4k

= 5i + 4j – 5k.

(i) Find the angle between AB and AC.[6]

(ii) Find the area of triangle ABC.[2]

Exam Questions

AB ACAB AC

cos .=

Exam Question

(ii)

The position vectors of three points A, B and C relative to an origin O are given respectively by

= 3i + 2j – 3k,

= 2i + j – 4k

= i + 4j – k.

Find the angle between AB and AC.

On w/bs

AB ACAB AC

cos .=

2. Lines L1, L2 and L3 have vector equations

L1: r = (5i – j – 2k) + s(–6i + 8j – 2k),

L2: r = (3i – 8j) + t( i + 3j + 2k),

L3: r = (2i + j + 3k) + u(3i + cj + k).

(i) Calculate the acute angle between L1 and L2.[4]

(ii) Given that L1 and L3 are parallel, find the value of c.[2]

Finding the angle between two vectors linesTo find the angle between two vector lines we only need consider the direction of the lines.

Re-call: r = a + t(b – a)

Let p = and q =

Finding the angle between two vectors linesSo the angle between them will be

||||.cos

qpqp

(ii) If the lines are parallel then they must have the same direction.

Do exercise 9G page 127. Qu 2,4,6 and exercise 9H page 131. Qu’s 3, 4 & 5.

Lines L1 and L2 have vector equations

L1: r = (i – j – 9k) + s(–6i + 8j – 4k),

L2: r = (2i – 8j) + t( 3i + 2j + 1k),

Calculate the acute angle between L1 and L2.

On w/b

Do exercise 9G page 127. Qu 2,4,6 and exercise 9H page 131. Qu’s 3, 4 & 5.

Also Moodle h/w parametric equations. For test next Monday. –Can you differentiate sin4x?