Question

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Answer 1

Correct Question :

Find the unit vector in the direction of the sum of the vectors a = 2i + 2j - 5k and b = 2i + j + 3k

Answer:

The required unit vector is   $\bf \hat{n}=\dfrac{4}{\sqrt{29}} \hat{i}+\dfrac{3}{\sqrt{29}} \hat{j}-\dfrac{2}{\sqrt{29}} \hat{k}$

Explanation:

Let c be the sum of the vectors a and b

Sum of the vectors :

 c = a + b

 $\sf c = 2\hat{i} + 2\hat{j} - 5\hat{k} + 2\hat{i} +\hat{j} + 3\hat{k} \\\\ \sf c=2\hat{i} + 2\hat{i} +2\hat{j}+\hat{j} - 5\hat{k} + 3\hat{k} \\\\ \sf c=4\hat{i}+3\hat{j}-2\hat{k}$

Magnitude of the vector c :

| c | = | a + b |

| c | = √(4² + 3² + (-2)²)

| c | = √(16 + 9 + 4)

| c | = √29

The required unit vector :

   $\sf \hat{n}=\dfrac{\overrightarrow{a} + \overrightarrow{b} }{|\overrightarrow{a}+\overrightarrow{b}|} \\\\\\ \sf \hat{n}=\dfrac{\overrightarrow{c}}{|\overrightarrow{c}|} \\\\\\ \sf \hat{n}=\dfrac{4\hat{i}+3\hat{j}-2\hat{k}}{\sqrt{29}} \\\\\\ \sf \hat{n}=\dfrac{4}{\sqrt{29}} \hat{i}+\dfrac{3}{\sqrt{29}} \hat{j}-\dfrac{2}{\sqrt{29}} \hat{k}$

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Unit vector :

• A vector that has magnitude of 1 unit.

• Also known as Direction vector.

• Unit vector = vector/magnitude of the vector