For the vectors in Fig. 3-32, witha=4,b=3, andc=5, calculate (a)a.b, (b)a-c, and (c)b.c

Short Answer

Expert verified

(a) The dot product a.b, is 0 .

(b) The dot product a-cis equal to -16 units.

(c) The dot productb.c is equal to -9 units.

Step by step solution

01

Given data

a=4,b=3andc=5.

02

Understanding the concept

Use the formula of scalar product of two vectors to find their products. To find the angle between the vectors, use the given diagram and trigonometry.

The formula for the dot product is,

a.b=abcosθ..............1

03

(a) Calculate the scalar product a→.b→ 

From the figure, it is seen that the angle between aandbis90° . Use equation (i) to calculate the dot product.

role="math" localid="1658469445125" a.b=abcos90=0unit

As the cos90is 0, the dot product of aandb is equal to 0.

04

(b) Calculate the scalar product a→.c→

To find the angle between aand c, use the figure. From figure, it can be seen that the angle betweenaand cis 180-θ.

We know that,

cos180-θ=-cosθ

Now, use the value in equation (i) to find the dot product between aandc.

role="math" localid="1658469899162" a.c=accos180-θ=-accosθ=-4×5×45=-16units

Therefore, the dot product between aand cis -16 units.

05

(c) The scalar product of two vectors b→.c→

Use the concept used in part (b) to find the dot product ofbandc. From figure,

cosφ=35

Now, use equation (i) to write the dot product betweenband c.

b.c=bc.ccosττ-φ=-bc.cosθ=-3×5×35=-9units

Therefore, the dot product between band cis -9 units .

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