Question

The component of vector $2i+3j+2k$ perpendicular to $i+j+k$ is:
A) ${\displaystyle \frac{5}{3}}\left(i-2j+k\right)$
B) ${\displaystyle \frac{1}{3}}\left(5i+j-2k\right)$
C) ${\displaystyle \frac{\left(7i-10j+7k\right)}{3}}$
D) ${\displaystyle \frac{5i-8j+5k}{3}}$

Answer 1

Hint: A vector quantity is such a quantity that has both magnitude as well as direction as opposed to a scalar quantity which only has a magnitude. For performing calculations with vector quantities a separate branch of mathematics known as vector algebra was formed. Vector algebra deals with the algebraic operations like addition, subtraction, multiplication etc. of vector quantities.

Complete step by step answer:
  Letus consider that we have been provided with two vectors a and b such that,
$\overrightarrow{a}=2i+3j+2k$
$\overrightarrow{b}=i+\text{j}+\text{k}$
We know that the component of vector a perpendicular to vector b can be obtained by the following expression.
$\overrightarrow{c}=\overrightarrow{a}-{\displaystyle \frac{\overrightarrow{a}\cdot \overrightarrow{b}}{{\left|\overrightarrow{b}\right|}^2}}\times \overrightarrow{b}$ …….(1)
Where, vector c is the component of vector a perpendicular to the vector b.
The magnitude of vector a is,
$\left|\overrightarrow{a}\right|=\sqrt{2^2+3^2+2^2}=\sqrt{17}$
The magnitude of vector b is,
$\left|\overrightarrow{b}\right|=\sqrt{1^2+1^2+1^2}=\sqrt{3}$.....(2)
The scalar or dot product of vectors a & b is given by,
$\overrightarrow{a}\cdot \overrightarrow{b}=2(1)-3(1)+2(1)=1$......(3)
Now, putting all the values from equations (2) & (3) in equation (1) we get,
$\overrightarrow{c}=2i+3j+2k-{\displaystyle \frac{1}{{\left(\sqrt{3}\right)}^2}}\times \left(i+\text{j}+\text{k}\right)$
$\overrightarrow{c}={\displaystyle \frac{5}{3}}\left(i-2j+k\right)$
i.e. ${\displaystyle \frac{5}{3}}\left(i-2j+k\right)$ is the vector which is the component of vector a and also perpendicular to vector b.

Hence option (A) is the correct answer option.

Note: For a vector quantity q $\overrightarrow{q}=ai+bj+ck$ a, b and c are the magnitudes of the quantity along x, y and z directions respectively. i is the unit vector along x - direction, j is the unit vector along y - direction, k is the unit vector along z - direction. So if a $\overrightarrow{q}$ is a force vector and it is given in Newton, then it means that a Newton of force is applied in x - direction, b Newton Of force is applied in y - direction and c Newton of force is acting in y - direction.