Question

How do we prove the projection law using vectors?

Answer 1

Hint: For this question, first you need to know the formula of vector projections. Then you need to write it down and write the formula of the unit vector. Then you have to multiply both the numerator and the denominator with b vectors. Then you can use the formula of the dot product of two vectors and finally substitute it in the equation to get the answer.

Complete step by step solution:
According to the problem, we are asked to prove the projection law using vectors
For this, first we need to write the vector projection of one vector on another vector. Here we will consider a as one vector and b as another vector. We consider this equation as equation 1.
Vector projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ is given by
Projection = $\left|\overrightarrow{a}\right|\cos \theta$ . unit vector of $\overrightarrow{b}$ ------- (1)
The unit vector of b can be written as ${\displaystyle \frac{\overrightarrow{b}}{\left|\overrightarrow{b}\right|}}$. Therefore substituting this in equation 1, we
 get:
$\Rightarrow projection=\left|\overrightarrow{a}\right|\cos \theta .{\displaystyle \frac{\overrightarrow{b}}{\left|\overrightarrow{b}\right|}}$
Now, we will multiply both the numerator and denominator with the $\left|\overrightarrow{b}\right|$. Therefore, by doing this we get:
$\Rightarrow projection=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta .{\displaystyle \frac{\overrightarrow{b}}{{\left|\overrightarrow{b}\right|}^2}}$ ------- (2)
$\overrightarrow{a}.\overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta$. Putting this in the above equation we get the following:
$\Rightarrow projection=(\overrightarrow{a}.\overrightarrow{b}){\displaystyle \frac{\overrightarrow{b}}{{\left|\overrightarrow{b}\right|}^2}}$
Therefore, we proved the projection law for two vectors using vectors.

Note: It is important that you should know the basic formulas for vectors. To solve this problem, you have to know the projection of one vector on another vector formula and you need to know the dot product of two vectors formulas.