Exams Joint Entrance Examination Mathematics Vector Algebra Question 78 Question 78 of 379 Medium MCQ
Let a ⃗ = 2 i ^ + 7 j ^ − k ^ , b ⃗ = 3 i ^ + 5 k ^ \vec{a}=2\hat{i}+7\hat{j}-\hat{k},\vec{b}=3\hat{i}+5\hat{k} a = 2 i ^ + 7 j ^ − k ^ , b = 3 i ^ + 5 k ^ and c ⃗ = i ^ − j ^ + 2 k ^ \vec{c}=\hat{i}-\hat{j}+2\hat{k} c = i ^ − j ^ + 2 k ^ . Let d ⃗ \vec{d} d be a vector which is perpendicular to both a ⃗ \vec{a} a and b ⃗ \vec{b} b , and c ⃗ ⋅ d ⃗ = 12 \vec{c}\cdot \vec{d}=12 c ⋅ d = 12 . Then ( − i ^ + j ^ − k ^ ) ⋅ ( c ⃗ × d ⃗ ) (-\hat{i}+\hat{j}-\hat{k})\cdot (\vec{c}\times \vec{d}) ( − i ^ + j ^ − k ^ ) ⋅ ( c × d ) is equal to :
Solutions · 1 Post your solution Official solution
If d ⃗ \vec{d} d is ⊥ \perp ⊥ to both a ⃗ \vec{a} a and b ⃗ \vec{b} b then
d ⃗ = λ ( a ⃗ × b ⃗ ) = λ ∣ i ^ j ^ k ^ 2 7 − 1 3 0 5 ∣ = ( 35 i ^ − 13 j ^ − 21 k ^ ) λ \vec{d}=\lambda (\vec{a}\times \vec{b})=\lambda \begin{vmatrix} \hat{i} \quad \hat{j} \quad \hat{k} \\ 2 \quad 7 \quad -1 \\ 3 \quad 0 \quad 5 \end{vmatrix}=(35\hat{i}-13\hat{j}-21\hat{k})\lambda d = λ ( a × b ) = λ i ^ j ^ k ^ 2 7 − 1 3 0 5 = ( 35 i ^ − 13 j ^ − 21 k ^ ) λ
c ⃗ ⋅ d ⃗ = 12 ⇒ λ ( 35 × 1 + 13 × 1 − 21 × 2 ) = 12 ⇒ λ ( 6 ) = 12 ⇒ λ = 2 λ ⃗ = 2 ( 35 i ^ − 13 j ^ − 21 k ^ ) \begin{gathered} \vec{c}\cdot \vec{d}=12\Rightarrow \lambda (35\times 1+13\times 1-21\times 2)=12 \\ \Rightarrow \lambda (6)=12\Rightarrow \lambda =2 \\ \vec{\lambda }=2(35\hat{i}-13\hat{j}-21\hat{k}) \end{gathered} c ⋅ d = 12 ⇒ λ ( 35 × 1 + 13 × 1 − 21 × 2 ) = 12 ⇒ λ ( 6 ) = 12 ⇒ λ = 2 λ = 2 ( 35 i ^ − 13 j ^ − 21 k ^ )
( − i ^ + j ^ − k ^ ) ⋅ ( c ⃗ × d ⃗ ) = ∣ − 1 1 − 1 1 − 1 2 70 − 26 − 42 ∣ = − 94 + 182 − 44 = 44 \begin{gathered} (-\hat{i}+\hat{j}-\hat{k})\cdot (\vec{c}\times \vec{d}) \\ =|\begin{array}{|c|}-1 \quad 1 \quad -1 \\ 1 \quad -1 \quad 2 \\ 70 \quad -26 \quad -42\end{array}| \\ =-94+182-44=44 \end{gathered} ( − i ^ + j ^ − k ^ ) ⋅ ( c × d ) = ∣ − 1 1 − 1 1 − 1 2 70 − 26 − 42 ∣ = − 94 + 182 − 44 = 44 Answer the question to unlock