Question

If $\overrightarrow A = 2\widehat i + 7\widehat j + 3\widehat k$ and $\overrightarrow B = 3\widehat i + 2\widehat j + 5\widehat k$, then find the projection of $\overrightarrow a$ and $\overrightarrow b$.

Answer 1

The problem has a vector $\overrightarrow A$ projected on vector $\overrightarrow B$. Thus, one vector component of $\overrightarrow A$ is parallel to the vector $\overrightarrow B$. We can easily calculate the projection of $\overrightarrow A$ and $\overrightarrow B$ by using one formula
${\text{Pro}}{{\text{j}}_{(\overrightarrow {B)} }}^{(\overrightarrow {A)} } = \,\dfrac{1}{{|\overrightarrow B |\,}}\,(\overrightarrow A \,.\,\overrightarrow B )\,\,or\,\,\dfrac{{(\overrightarrow A \,.\,\overrightarrow B )\,}}{{|\overrightarrow B |\,}}$

Stepwise solution:

Given:
$\overrightarrow A = 2\widehat i + 7\widehat j + 3\widehat k$
$\overrightarrow B = 3\widehat i + 2\widehat j + 5\widehat k$
If it is given in the question that $\overrightarrow A$ projects on $\overrightarrow B$ , so that the vector component of $\overrightarrow A$ is in the direction of vector component $\overrightarrow B$. The orthogonal projection of A onto a straight line parallel to B, which is defined as $A = A\widehat B$ where A is scalar, called a scalar projection of A onto B, and B is the unit vector in the direction of B.
Thus, the projection of $\overrightarrow A$ on $\overrightarrow B$ can be written as:
$\Pr o{j_{(\overrightarrow {B)} }}^{(\overrightarrow {A)} } = \,\dfrac{1}{{|\overrightarrow B |\,}}\,(\overrightarrow A \,.\,\overrightarrow B )\,$ eq. (1)
Where, $(\overrightarrow A \,.\,\overrightarrow B )\,$is the dot product of two vectors $\overrightarrow A$and $\overrightarrow B$ and $|\overrightarrow B |$ is the magnitude of vector $\overrightarrow B$.
$\overrightarrow A = 2\widehat i + 7\widehat j + 3\widehat k$ and $\overrightarrow B = 3\widehat i + 2\widehat j + 5\widehat k$
So, dot product of $\overrightarrow A$ and $\overrightarrow B$ gives:
$(\overrightarrow A .\overrightarrow B )\, = \,(2\widehat i + 7\widehat j + 3\widehat k).\,(3\widehat i + 2\widehat j + 5\widehat k)$
$\Rightarrow (2 \times 3) + (7 \times 2) + (3 \times 5)$ eq. (2)
$\Rightarrow (\overrightarrow A .\,\overrightarrow B )\, = \,35$
Here, we take one product of the numbers in the same co ordinate and then add them all to obtain $(\overrightarrow A .\,\overrightarrow B )\,$
Also, remember that the dot product of $\widehat i$, $\widehat j$ and $\widehat k$ with itself is 1.
Now,
$|\overrightarrow B |\, = \,\sqrt {{{(\widehat i)}^2}\, + \,{{(\widehat j\,)}^2} + \,{{(\widehat k)}^2}}$
$|\overrightarrow B |\, = \,\sqrt {{{(3)}^2}\, + \,{{(2\,)}^2} + \,{{(5)}^2}}$
$|\overrightarrow B |\, = \,\sqrt {38}$
Now putting the value in eq. (1), we get
$\Pr o{j_{(\overrightarrow {B)} }}^{(\overrightarrow {A)} } = \,\dfrac{{35}}{{\sqrt {38} }} = \,5.67$

Note:
Students must have an idea of certain vector properties before attempting this question. The scalar products of two vectors a and b, denoted by $\overrightarrow a .\,\overrightarrow b$ is defined as $\overrightarrow a .\,\overrightarrow b = |\,\overrightarrow a |\,|\overrightarrow b |\,\cos \,\theta$ where $\theta$ is the angle between vector and vector b, $0 \leqslant \theta \leqslant \pi$. If either vector (a) equal to zero or vector (b) equal to zero, then $\theta$ is not defined, and in this case, we define $\overrightarrow a .\,\overrightarrow b = 0$.