Question

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

The scalar projection of B on A is -1 .

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

Step-by-step solution

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 $\mathit{\theta }$between Aand B.

Calculation of scalar projection:

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

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

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

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

Let the angle between the vectors A and B be $\theta$

$$\begin{gathered} A.B=AB\cos \theta \to \cos \theta \\ =\frac{A.B}{AB} \end{gathered}$$

$$\begin{gathered} \cos \theta =\frac{A.B}{AB} \\ =\frac{-5}{5\times 3} \\ =\frac{-1}{3} \end{gathered}$$

Hence, the scalar projection of B on A is -1 . And the scalar projection of A on B is $\frac{-5}{3}$.