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Chapter 3: Q13P (page 105)

If A=4i-3k and B=-2i+2j-k,find the scalar projection of A on B the scalar projection of B on A,and the cosine of the angle between A and B .

Short Answer

Expert verified

The scalar projection of B on A is -1 .

The scalar projection of A on B is $\frac{- 5}{3}$ .

Step by step solution

01

Concept of scalar:

The scalar product of A and B written A.Bis a scalar equal to the magnitude of A times the magnitude of B times the cosine of the angle $胃$ between Aand B.

02

Calculation of scalar projection:

The scalar projection of B on A is given by $\frac{A . B}{A}$ .

$\frac{A . B}{A} = \frac{(4) (- 2) + (0) (2) + (- 3) (- 1)}{\sqrt{4^{2} + 0^{2} + {(- 3)}^{2}}} = \frac{- 5}{5} = - 1$

The scalar projection of A on B is given by $\frac{A . B}{A}$ .

$\frac{A . B}{A} = \frac{(4) (- 2) + (0) (2) + (- 3) (- 1)}{\sqrt{{(- 2)}^{2} + 2^{2} + {(- 1)}^{2}}} = \frac{- 5}{3}$

Let the angle between the vectors A and B be $胃$

$A . B = A B \cos 胃鈫� \cos 胃 = \frac{A . B}{A B}$

$\cos 胃 = \frac{A . B}{A B} = \frac{- 5}{5 脳 3} = \frac{- 1}{3}$

Hence, the scalar projection of B on A is -1 . And the scalar projection of A on B is

\frac{- 5}{3}

.

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