Find the angle between the line r=(i^+j^−2k^)+λ(i^−j^+k^) and the plane r⋅(2i^−j^+k^)=4.

Given line is r=a+λb and given plane is r⋅n=psinθ=∣b∣∣n∣∣b⋅n∣={12+(−1)2+12}{22+(−1)2+12}∣(i^−j^+k^)⋅(2i^−j^+k^)∣=(3×6)∣(2×1)+(−1)×(−1)+1×1∣=184=(324×22)=322⇒θ=sin−1(322).

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Find the angle between the line {r}=({i}+{j}-2{k})+({i}-{j}+{k

Question

Find the angle between the line $r = (\hat{i} + \hat{j} - 2 \hat{k}) + λ (\hat{i} - \hat{j} + \hat{k})$ and the plane $r \cdot (2 \hat{i} - \hat{j} + \hat{k}) = 4$ .

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Sebastian Discussed

Find the angle between the line

r = (\hat{i} + \hat{j} - 2 \hat{k}) + λ (\hat{i} - \hat{j} + \hat{k})

and the plane

r \cdot (2 \hat{i} - \hat{j} + \hat{k}) = 4

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10 mins ago

Text solutionVerified

Given line is

r = a + λ b

and given plane is

r \cdot n = p

sin θ = \frac{∣ b \cdot n ∣}{∣ b ∣∣ n ∣} = \frac{∣ ( i ^ - j ^ + k ^ ) \cdot ( 2 i ^ - j ^ + k ^ ) ∣}{{ 1 ^{2} + ( - 1 ) ^{2} + 1 ^{2} } { 2 ^{2} + ( - 1 ) ^{2} + 1 ^{2} }}

= \frac{∣ ( 2 \times 1 ) + ( - 1 ) \times ( - 1 ) + 1 \times 1∣}{( 3 \times 6 )}

= \frac{4}{18} = (\frac{4}{3 2} \times \frac{2}{2}) = \frac{2 2}{3}

\Rightarrow θ = sin^{- 1} (\frac{2 2}{3})

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Question Text	Find the angle between the line $r = (\hat{i} + \hat{j} - 2 \hat{k}) + λ (\hat{i} - \hat{j} + \hat{k})$ and the plane $r \cdot (2 \hat{i} - \hat{j} + \hat{k}) = 4$ .
Answer Type	Text solution:1
Upvotes	150

Find the angle between the line r=(i^+j^​−2k^)+λ(i^−j^​+k^) and the plane r⋅(2i^−j^​+k^)=4.

Text solutionVerified

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Find the angle between the line $r = (\hat{i} + \hat{j} - 2 \hat{k}) + λ (\hat{i} - \hat{j} + \hat{k})$ and the plane $r \cdot (2 \hat{i} - \hat{j} + \hat{k}) = 4$ .