Question

If it is given that vectors $\overrightarrow{a}=x\hat{i}+2\hat{j}-z\hat{k}$ and $\overrightarrow{b}=3\hat{i}-y\hat{j}+\hat{k}$ are two equal vectors, then write the value of x + y + z.

Answer 1

Hint: To solve this question, we will first define what a vector is and its representation in 3D space. Then we will define what are equal vectors. We will apply the condition for equal vectors on the given vectors and try to find the value of x, y, and z. Once we get all the values, we can add them to get x + y + z.

Complete step-by-step solution:
A vector is an entity in 3D space which has magnitude as well as direction. It is used to represent physical phenomena as force, acceleration, velocity, etc. In 3D space, it is represented by a ray.
A figure of a vector in 3D space is as follows:

For two vectors to be equal, they should be equal in magnitude as well as the direction. Hence, the coefficient of $\hat{i}$, $\hat{j}$ and $\hat{k}$ must be equal in two equal vectors.
We are given that vectors $\overrightarrow{a}$ is equal to vector $\overrightarrow{b}$, where $\overrightarrow{a}=x\hat{i}+2\hat{j}-z\hat{k}$ and $\overrightarrow{b}=3\hat{i}-y\hat{j}+\hat{k}$.
$\Rightarrow x\hat{i}+2\hat{j}-z\hat{k}=3\hat{i}-y\hat{j}+\hat{k}$
The coefficient of $\hat{i}$ on the left-hand side is x, whereas on the right-hand side is 3.
$\Rightarrow $ x = 3
The coefficient of $\hat{j}$ on the left-hand side is 2, whereas on the right-hand side is ─y.
$\Rightarrow $ y = ─2
The coefficient of $\hat{z}$ on the left-hand side is ─z, whereas on the right-hand side is 1.
$\Rightarrow $ z = ─1
Therefore, x + y + z = 3 – 2 – 1
Hence, x + y + z = 0.

Note: A vector is generally denoted as $x\hat{i}+y\hat{j}+z\hat{k}$, where x, y, and z are the coordinate of its tip in the 3D space and $\hat{i}$, $\hat{j}$ and $\hat{k}$ are unit vectors in the direction x, y, and z respectively. The magnitude of the vector is $\sqrt{{{x}^{2}}+{{y}^{2}}+{{z}^{2}}}$ and the direction is of the vector is from origin (0, 0, 0) to the tip (x, y, z).