Question 13 of 68intermediate🔧 ApplyNumerical2 marks

Find a vector in the direction of vector 5î - ĵ + 2k̂ which has magnitude 8 units.

Correct Answer

The required vector is 830\frac{8}{\sqrt{30}}(5i^−j^+2k^\hat{i} - \hat{j} + 2 \hat{k}) or 4030i^−830j^+1630k^\frac{40}{\sqrt{30}} \hat{i} - \frac{8}{\sqrt{30}} \hat{j} + \frac{16}{\sqrt{30}} \hat{k}

Exercise: EXERCISE 10.2 | Q: 10 | (Page 17)
For More Understanding

Explanation

This question follows the same pattern as Example 7 in the textbook. To find a vector with a specific magnitude in a given direction, we first find the unit vector in that direction, then multiply it by the required magnitude. The unit vector formula is a^=1∣a⃗∣a⃗\hat{a} = \frac{1}{|\vec{a}|} \vec{a}.

Solution Steps

  1. Step 1: Let the given vector be a⃗=5i^−j^+2k^\vec{a} = 5 \hat{i} - \hat{j} + 2 \hat{k}

  2. Step 2: Find the magnitude: |a⃗\vec{a}| = 52+(−1)2+22=25+1+4=30\sqrt{5^2 + (-1)^2 + 2^2} = \sqrt{25 + 1 + 4} = \sqrt{30}

  3. Step 3: Find the unit vector in the direction of a⃗\vec{a}: a^=130\hat{a} = \frac{1}{\sqrt{30}}(5i^−j^+2k^\hat{i} - \hat{j} + 2 \hat{k})

  4. Step 4: Multiply the unit vector by the required magnitude 8: Required vector = 8a^=830\hat{a} = \frac{8}{\sqrt{30}}(5i^−j^+2k^\hat{i} - \hat{j} + 2 \hat{k}) = 4030i^−830j^+1630k^\frac{40}{\sqrt{30}} \hat{i} - \frac{8}{\sqrt{30}} \hat{j} + \frac{16}{\sqrt{30}} \hat{k}

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