Question

Find the unit vector in the direction of vector A=4i +3j

Answer 1

Answer:

$\displaystyle \vec u_A=\frac{4}{5}\hat i +\frac{3}{5}\hat j$

Explanation:

Unit Vector

The unit vector associated with a given vector $\vec a$ is another vector pointing in the same direction of $\vec a$ and with magnitude 1.

The unit vector can be calculated as follows:

$\displaystyle \vec u_a=\frac{\vec a }{||\vec a ||}$

Where $||\vec a ||$ is the magnitude of the vector.

If $\vec a$ is given as:

$\vec a = x\hat i +y\hat j$

The magnitude of the vector is:

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

We have:

$\vec A=4\hat i +3\hat j$

${||\vec a ||}=\sqrt{4^2+3^2}=\sqrt{25}=5$

Thus the unit vector is:

$\displaystyle \vec u_A=\frac{4\hat i +3\hat j }{5}$

Simplifying:

$\mathbf{\displaystyle \vec u_A=\frac{4}{5}\hat i +\frac{3}{5}\hat j}$