Question 55 of 68intermediate🔧 ApplyNumerical2 marks

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors a⃗ = 2î + 3ĵ - k̂ and b⃗ = î - 2ĵ + k̂.

Correct Answer

The required vector is 1510i^+510j^\frac{15}{\sqrt{10}} \hat{i} + \frac{5}{\sqrt{10}} \hat{j}

Exercise: MISCELLANEOUS EXERCISE ON CHAPTER 10 | Q: 6 | (Page 35)
For More Understanding

Explanation

This question combines two concepts from the textbook: finding the resultant (sum) of two vectors (as shown in Example 8), and finding a vector of given magnitude in a specific direction (as shown in Example 7). The student must first compute the resultant vector, then find its unit vector, and finally multiply by 5 to get the vector of magnitude 5.

Solution Steps

  1. Step 1: Find the resultant vector a⃗+b⃗\vec{a} + \vec{b} = (2i^+3j^−k^\hat{i} + 3 \hat{j} - \hat{k}) + (i^−2j^+k^\hat{i} - 2 \hat{j} + \hat{k}) = 3i^+j^\hat{i} + \hat{j}

  2. Step 2: Find the magnitude of the resultant |a⃗+b⃗\vec{a} + \vec{b}| = 32+12=10\sqrt{3^2 + 1^2} = \sqrt{10}

  3. Step 3: Find the unit vector in the direction of resultant c^=110\hat{c} = \frac{1}{\sqrt{10}}(3i^+j^\hat{i} + \hat{j}) = 310i^+110j^\frac{3}{\sqrt{10}} \hat{i} + \frac{1}{\sqrt{10}} \hat{j}

  4. Step 4: Multiply unit vector by 5 to get required vector Required vector = 5c^=5(310i^+110j^)=1510i^+510j^\hat{c} = 5 \left(\frac{3}{\sqrt{10}}\hat{i} + \frac{1}{\sqrt{10}}\hat{j}\right) = \frac{15}{\sqrt{10}} \hat{i} + \frac{5}{\sqrt{10}} \hat{j}

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