Question

If $\vec{A}=2\hat{i}+7\hat{j}+3\hat{k}$ and $\vec{B}=3\hat{i}+2\hat{j}+5\hat{k}$, find the component of $\vec{A}$ along $\vec{B}$.

Answer 1

Hint: The projection of a vector on another vector gives the component along that vector. So find the projection of vector A along vector B.
Projection of a vector A along another vector B is its dot product with unit vector along vector B.

Complete answer:
The projection of a vector on another vector gives the component along that vector. Projection of a vector A along another vector B is its dot product with unit vector along vector B.
We have $\vec{A}=2\hat{i}+7\hat{j}+3\hat{k}$ and $\vec{B}=3\hat{i}+2\hat{j}+5\hat{k}$. We first find the unit vector along vector B.
Unit vector is defined as a vector with unit magnitude. It can be determined as a ratio of vectors to its magnitude. Therefore, unit vector along vector B can be written as
$\hat{B}=\dfrac{\mathbf{B}}{\left| \mathbf{B} \right|}$
Magnitude of vector B is
$\left| \mathbf{B} \right|=\sqrt{{{3}^{2}}+{{2}^{2}}+{{5}^{2}}}=\sqrt{38}$
Therefore we have
$\hat{B}=\dfrac{{\vec{B}}}{B}=\dfrac{1}{\sqrt{38}}\left( 3\hat{i}+2\hat{j}+5\hat{k} \right)$
Dot product of two vectors is defined as a product of corresponding components of the vectors. Therefore, dot product of $\vec{A}$ and $\hat{B}$ is
$\vec{A}.\hat{B}=\dfrac{1}{\sqrt{38}}\left( 2\hat{i}+7\hat{j}+3\hat{k} \right)\left( 3\hat{i}+2\hat{j}+5\hat{k} \right)$
$\mathbf{A}.\hat{B}=\dfrac{1}{\sqrt{38}}(2\times 3+7\times 2+3\times 5)=\dfrac{35}{\sqrt{38}}$
The vector-form of this projection is given as
$\left( \mathbf{A}.\hat{B} \right)\hat{B}=\dfrac{35}{\sqrt{38}}\hat{B}=\dfrac{35}{\sqrt{38}}\dfrac{1}{\sqrt{38}}\left( 3\hat{i}+2\hat{j}+5\hat{k} \right)=\dfrac{35}{38}\left( 3\hat{i}+2\hat{j}+5\hat{k} \right)$
This form can be simplified if required.

Additional Information:
Those physical quantities which have only magnitude and not direction are known as scalars. Some examples are mass, density, speed, distance and work.
The quantities which require not only magnitude but direction also to complete their representation are known as vector quantities. Some examples are displacement, velocity and weight.

Note:
The projection of a vector on another vector gives the component along that vector. The component of a vector is a scalar quantity and can be converted to a vector component by multiplying with a unit vector.