Question

How do you find the unit vector in the same direction of the given vector $\overrightarrow{a}=(-10,6,-7)$?

Answer 1

Hint: We start solving the problem by making use of the result that the magnitude of the vector $(x,y,z)$ is $\sqrt{x^2+y^2+z^2}$ to find the magnitude of the given vector $\overrightarrow{a}=(-10,6,-7)$. We then make use of the fact that the unit vector in the direction of the vector $\overrightarrow{r}=(x,y,z)$ is defined as $${\displaystyle \frac{\overrightarrow{r}}{\left|\overrightarrow{r}\right|}}$$. We use this result to find the required unit vector of the given vector $\overrightarrow{a}=(-10,6,-7)$.

Complete step by step answer:
According to the problem, we are asked to find the unit vector in the same direction of the given vector $\overrightarrow{a}=(-10,6,-7)$.
Let us find the magnitude of the given vector $\overrightarrow{a}$.
We know that the magnitude of the vector $(x,y,z)$ is $\sqrt{x^2+y^2+z^2}$. Let us use this result to find the magnitude of the given vector $\overrightarrow{a}$.
So, the magnitude of the given vector $\overrightarrow{a}$ is $\left|\overrightarrow{a}\right|=\sqrt{{(-10)}^2+6^2+{(-7)}^2}=\sqrt{100+36+49}=\sqrt{185}$ ---(1).
We know that the unit vector in the direction of the vector $\overrightarrow{r}=(x,y,z)$ is defined as $${\displaystyle \frac{\overrightarrow{r}}{\left|\overrightarrow{r}\right|}}$$. Using this result, we get the unit vector in the direction of vector $\overrightarrow{a}=(-10,6,-7)$ as $${\displaystyle \frac{\overrightarrow{a}}{\left|\overrightarrow{a}\right|}}$$.
From equation (1), we get the unit vector as ${\displaystyle \frac{\overrightarrow{a}}{\left|\overrightarrow{a}\right|}}={\displaystyle \frac{(-10,6,-7)}{\sqrt{185}}}$.

$\therefore$ We have found the unit vector in the direction of given vector $\overrightarrow{a}=(-10,6,-7)$ as ${\displaystyle \frac{(-10,6,-7)}{\sqrt{185}}}$.

Note: We can verify the obtained result by finding the magnitude of the obtained vector as the magnitude of the unit vectors is 1 and the angle of the obtained unit vector with given vector should be 0 as both are in the same direction. We can also solve the problem by making use of the fact that the vector in the direction of the vector $\overrightarrow{x}$ is $\lambda \overrightarrow{x}$ and then equating its magnitude to 1. Similarly, we can expect problems to find the unit vector opposite to the given vector $\overrightarrow{a}=(-10,6,-7)$.