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

2 Answers
Jan 27, 2018

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^@#

Jan 27, 2018

#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.

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