Question

find the unit vector in the direction of vector 4i- 4j+2k​

Answer 1

Explanation:

Hope the answer is satisfing

Answer 2

The unit vector in the direction of the given vector will be $\hat p = \frac{2}{3} \hat i - \frac{2}{3} \hat j +\frac{1}{3} \hat k}$.

Given,

A vector:

4i- 4j+2k​

To find,

The unit vector, in the direction of the given vector.

Solution,

Let there be a vector P, such that,

$\vec P=x \hat i + y \hat j + z \hat k$

And the unit vector in the direction of $\vec P$ be, $\hat a$.

Now, the unit vector $\hat a$ is given as,

$\hat a = \frac{\vec P}{| \vec P |}$

Where $|\vec P|$ is the magnitude of vector P, and is given by,

$|\vec P|=\sqrt{x^2 + y^2 + z^2}$

So, for the given vector, (say $\vec P$), let the unit vector be $\hat p$.

Now, $|\vec P|$ will be,

$|\vec P| = \sqrt{(4)^2 + (-4)^2 + (2)^2}$

$\implies |\vec P| = \sqrt{16 + 16 + 4}=\sqrt{36}$

$\implies |\vec P| = 6$

Since,

$\vec P=4 \hat i -4 \hat j + 2 \hat k$

The unit vector $\hat p$ will be,

$\hat p = \frac{\vec P}{| \vec P|}$

$\implies \hat p = \frac{4 \hat i -4 \hat j + 2 \hat k}{6}$

$\implies \hat p = \frac{2}{3} \hat i - \frac{2}{3} \hat j +\frac{1}{3} \hat k}$

Therefore, the unit vector in the direction of the given vector will be $\hat p = \frac{2}{3} \hat i - \frac{2}{3} \hat j +\frac{1}{3} \hat k}$.