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CBSE Class 12 Math 2008 Solved Paper

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Question : 9 of 29
Marks: +1, -0
Find the angle between the vectors a⃗\vec{a} = i^−j^+k^\hat{i}-\hat{j}+\hat{k} and b⃗\vec{b} = i^+j^−k^\hat{i}+\hat{j}-\hat{k}
Solution:  
a⃗\vec{a} = i^−j^+k^\hat{i}-\hat{j}+\hat{k}
b⃗\vec{b} = i^+j^−k^\hat{i}+\hat{j}-\hat{k}
Now , a⃗.b⃗\vec{a}.\vec{b} = ∣a⃗∣ ∣b⃗∣cos⁡θ|\vec{a}|\,|\vec{b}|\cos\theta
Where θ is the angle between a⃗\vec{a} and b⃗\vec{b} ; also 0 ≤ θ ≤ π
i^−j^+k^\hat{i}-\hat{j}+\hat{k} , i^+j^−k^\hat{i}+\hat{j}-\hat{k} = (12+(−1)2+12)(\sqrt{1^2+(-1)^2+1^2}) (12+12+(−1)2)(\sqrt{1^2+1^2+(-1)^2}) cos θ
⇒ 1 - 1 - 1 = 3 cos θ
- 1 = 3 cos θ
cos θ = −13-\frac{1}{3} ⇒ θ = cos⁡−1(−13)\cos^{-1}\left(-\frac{1}{3}\right)
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