Question 12 of 68intermediate🔧 ApplyNumerical2 marks

For given vectors, a⃗ = 2î - ĵ + 2k̂ and b⃗ = -î + ĵ - k̂, find the unit vector in the direction of the vector a⃗ + b⃗.

Correct Answer

12i^+12k^\frac{1}{\sqrt{2}} \hat{i} + \frac{1}{\sqrt{2}} \hat{k}

Exercise: EXERCISE 10.2 | Q: 9 | (Page 17)
For More Understanding

Explanation

This question is solved by following the method shown in Example 8 of the provided context. First, we determine the vector sum a⃗+b⃗\vec{a} + \vec{b} by adding the corresponding components. Next, we calculate the magnitude of this resultant vector using the formula ∣c⃗∣=x2+y2+z2|\vec{c}| = \sqrt{x^2 + y^2 + z^2}, as demonstrated in Example 6. Finally, we find the unit vector by dividing the resultant vector by its magnitude, adhering to the definition a^=1∣a⃗∣a⃗\hat{a} = \frac{1}{|\vec{a}|}\vec{a} provided in the context.

Solution Steps

  1. Step 1: Find the sum of the vectors. Given a⃗=2i^−j^+2k^\vec{a} = 2\hat{i} - \hat{j} + 2\hat{k} and b⃗=−i^+j^−k^\vec{b} = -\hat{i} + \hat{j} - \hat{k}. Let c⃗=a⃗+b⃗\vec{c} = \vec{a} + \vec{b}. c⃗=(2−1)i^+(−1+1)j^+(2−1)k^=i^+0j^+k^=i^+k^\vec{c} = (2-1)\hat{i} + (-1+1)\hat{j} + (2-1)\hat{k} = \hat{i} + 0\hat{j} + \hat{k} = \hat{i} + \hat{k}.

  2. Step 2: Calculate the magnitude of the resultant vector. ∣c⃗∣=12+02+12=1+0+1=2|\vec{c}| = \sqrt{1^2 + 0^2 + 1^2} = \sqrt{1 + 0 + 1} = \sqrt{2}.

  3. Step 3: Find the unit vector in the direction of c⃗\vec{c}. The unit vector is given by c⃗∣c⃗∣\frac{\vec{c}}{|\vec{c}|}. c^=12(i^+k^)=12i^+12k^\hat{c} = \frac{1}{\sqrt{2}}(\hat{i} + \hat{k}) = \frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{k}.

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