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CBSE Class 12 Math 2025 All Sets Solved Paper

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Question : 2 of 20
Marks: +1, -0
The unit vector perpendicular to the vectors i^−j^\hat{i} - \hat{j} and i^+j^\hat{i} + \hat{j} is
Solution:  
a⃗=i^−j^\vec{a}=\hat{i}-\hat{j}
b⃗=i^+j^\vec{b}=\hat{i}+\hat{j}
Let n^\hat{n} be the unit vector perpendicular to both a⃗\vec{a} and b⃗\vec{b}
n^=a×b⃗⃗∣a×b⃗∣⃗\hat{n} = \frac{ \vec{a \times \vec{b}} }{ \vec{ | a \times \vec{b} | } }
a⃗×b⃗=(i^−j^)×(i^+j^)\vec{a} \times \vec{b} = (\hat{i} - \hat{j}) \times (\hat{i} + \hat{j})
=∣ij^k^1−10110∣^= \widehat{\begin{vmatrix} i & \hat{j} & \hat{k} \\ 1 & -1 & 0 \\ 1 & 1 & 0 \end{vmatrix}}
a⃗×b⃗=2k^\vec{a} \times \vec{b} = 2\hat{k}
∣a⃗×b⃗∣=2\left|\vec{a} \times \vec{b}\right| = 2
n^=2k^2=k^\hat{n} = \frac{ \widehat{2k} }{2} = \hat{k}
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