Question

If `A = <8 ,3 ,-7 >`, `B = <6 ,-9 ,5 >`, and `C=A-B`, what is the angle between A and C?

Answer 1

The angle is `=43.9^@`

Explanation:

Let's start by calculating

`vecC=vecA-vecB`

`vecC=〈8,3,-7〉-〈6,-9,5〉=〈2,12,-12〉`

The angle between `vecA` and `vecC` is given by the dot product definition.

`vecA.vecC=∥vecA∥*∥vecC∥costheta`

Where `theta` is the angle between `vecA` and `vecC`

The dot product is

`vecA.vecC=〈8,3,-7〉.〈2,12,-12〉=16+36+84=136`

The modulus of `vecA`= `∥〈8,3,-7〉∥=sqrt(64+9+49)=sqrt122`

The modulus of `vecC`= `∥〈2,12,-12〉∥=sqrt(4+144+144)=sqrt292`

So,

`costheta=(vecA.vecC)/(∥vecA∥*∥vecC∥)=136/(sqrt122*sqrt292)=0.72`

`theta=arccos(0.72)=43.9^@`

Answer 2

`43.9^@`

Explanation:

`A=[(8),(3),(-7)]`

`B=[(6),(-9),(5)]`

`C=A-B=[(8),(3),(-7)]-[(6),(-9),(5)]=[(8-6),(3-(-9)),(-7-5)]=[(2),(12),(-12)]`

We can find the angle between vectors using the Dot Product

The dot product states that for vectors a and b:

`color(blue)(a*b=||a||*||b||*cos(theta))`

The dot product is sometimes called the inner product, because of the way the vectors a and b are multiplied and summed.

We are used to multiplying brackets in the following way.

`(a+b)(c+d)=ac+ad+bc+bd`

In the dot product we multiply the vectors in the following way.

`(a+b+c) * (d+e+f)=ad+be+cf`

So we are multiplying corresponding components and then adding them together.

Let `a = [(x),(y),(z)]`

Magnitude of `a=||a||`

`color(blue)(||a||=sqrt(x^2+y^2+z^2))`

From our example:

First find the product of:

`A*C`

`[(8),(3),(-7)]*[(2),(12),(-12)]=[(8xx2),(3xx12),(-7xx-12)]`

`=[(16),(36),(84)]=16+36+84=136`

We now find the magnitudes of A and C:

`||A||=sqrt((8)^2+(3)^2+(-7)^2)=sqrt(122)`

`||C||=sqrt((2)^2+(12)^2+(-12)^2)=sqrt(292)=2sqrt(73)`

So we have for:

`a*b=||a||*||b||*cos(theta)`

`136=sqrt(122)*2sqrt(73)*cos(theta)`

`cos(theta)=136/(sqrt(122)*2sqrt(73))`

`theta=arccos(cos(theta))=arccos(136/(sqrt(122)*2sqrt(73)))=43.9^@`
( 2 .d.p.)

The angle between vectors A and C is `43.9^@`

From the diagram we can see that the angle found by the dot product, is the angle between the vectors where they are heading in the same direction.