Find the unit vector in the direction of the sum of the vectors v1 = i + 4j + 2k, v2 = 3i − 3j − 2k, and v3…

Question

Find the unit vector in the direction of the sum of the vectors v1 = i + 4j + 2k, v2 = 3i − 3j − 2k, and v3 = −2i + 2j + 6k.

ScanSolve AI Tutor · Accepted Answer

To find the unit vector in the direction of the sum of the given vectors, we first need to find the sum of the vectors and then divide it by its magnitude.

The given vectors are:
$\vec{v_1} = \hat{i} + 4\hat{j} + 2\hat{k}$
$\vec{v_2} = 3\hat{i} - 3\hat{j} - 2\hat{k}$
$\vec{v_3} = -2\hat{i} + 2\hat{j} + 6\hat{k}$

Find the sum of the vectors.
Let $\vec{R}$ be the resultant vector.
$\vec{R} = \vec{v_1} + \vec{v_2} + \vec{v_3}$ $\vec{R} = (\hat{i} + 4\hat{j} + 2\hat{k}) + (3\hat{i} - 3\hat{j} - 2\hat{k}) + (-2\hat{i} + 2\hat{j} + 6\hat{k})$
Combine the corresponding components:
$\vec{R} = (1+3-2)\hat{i} + (4-3+2)\hat{j} + (2-2+6)\hat{k}$ $\vec{R} = 2\hat{i} + 3\hat{j} + 6\hat{k}$
Calculate the magnitude of the resultant vector $\vec{R}$ .
The magnitude of a vector $a\hat{i} + b\hat{j} + c\hat{k}$ is $\sqrt{a^2 + b^2 + c^2}$ .
$|\vec{R}| = \sqrt{(2)^2 + (3)^2 + (6)^2}$ $|\vec{R}| = \sqrt{4 + 9 + 36}$ $|\vec{R}| = \sqrt{49}$ $|\vec{R}| = 7$
Find the unit vector in the direction of $\vec{R}$ .
The unit vector $\hat{R}$ is given by $\frac{\vec{R}}{|\vec{R}|}$ .
$\hat{R} = \frac{2\hat{i} + 3\hat{j} + 6\hat{k}}{7}$ $\hat{R} = \frac{2}{7}\hat{i} + \frac{3}{7}\hat{j} + \frac{6}{7}\hat{k}$
Comparing this result with the given options, it matches option (d).

The final answer is $\text{(d)}$ .

Find the unit vector in the direction of the sum of the vectors v1 = i + 4j + 2k, v2 = 3i − 3j − 2k, and v3 = -2i + 2j + 6k.

Answer

Step-by-step solution

Find the sum of the vectors.

Calculate the magnitude of the resultant vector $\vec{R}$ .

Find the unit vector in the direction of $\vec{R}$ .

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