Show that the line r=(2i^+5j^+7k^)+λ(i^+3j^+4k^) is parallel to the plane r⋅(i^+j^−k^)=7. Also, find the distance between them.

We know that a line r=a+λb is parallel to the plane r⋅n=q only when this line is perpendicular to the normal to the plane.So, we must have b⋅n=0Here, (b⋅n)=(i^+3j^+4k^)⋅(i^+j^−k^)=(1+3−4)=0Hence, the given line is parallel to the given plane.Required distance between the line and the plane=∣n∣∣a⋅n−q∣=∣i^+j^−k^∣∣(2i^+5j^+7k^)⋅(i^+j^−k^)−7∣=12+12+(−1)2∣2+5−7−7∣=3∣−7∣=37 units.

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Show that the line {r}=(2{i}+5{j}+7{k})+({i}+3{j}+4{k}) is par

Question

Show that the line $r = (2 \hat{i} + 5 \hat{j} + 7 \hat{k}) + λ (\hat{i} + 3 \hat{j} + 4 \hat{k})$ is parallel to the plane $r \cdot (\hat{i} + \hat{j} - \hat{k}) = 7$ . Also, find the distance between them.

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Show that the line

r = (2 \hat{i} + 5 \hat{j} + 7 \hat{k}) + λ (\hat{i} + 3 \hat{j} + 4 \hat{k})

is parallel to the plane

r \cdot (\hat{i} + \hat{j} - \hat{k}) = 7

. Also, find the distance between them.

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Text solutionVerified

We know that a line

r = a + λ b

is parallel to the plane

r \cdot n = q

only when

this line is perpendicular to the normal to the plane.

So, we must have

b \cdot n = 0

Here,

(b \cdot n) = (\hat{i} + 3 \hat{j} + 4 \hat{k}) \cdot (\hat{i} + \hat{j} - \hat{k}) = (1 + 3 - 4) = 0

Hence, the given line is parallel to the given plane.

Required distance between the line and the plane

= \frac{∣ a \cdot n - q ∣}{∣ n ∣} = \frac{∣ ( 2 i ^ + 5 j ^ + 7 k ^ ) \cdot ( i ^ + j ^ - k ^ ) - 7∣}{∣ i ^ + j ^ - k ^ ∣}

= \frac{∣2 + 5 - 7 - 7∣}{1 ^{2} + 1 ^{2} + ( - 1 ) ^{2}}

= \frac{∣ - 7∣}{3} = \frac{7}{3}

units.

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Question Text	Show that the line $r = (2 \hat{i} + 5 \hat{j} + 7 \hat{k}) + λ (\hat{i} + 3 \hat{j} + 4 \hat{k})$ is parallel to the plane $r \cdot (\hat{i} + \hat{j} - \hat{k}) = 7$ . Also, find the distance between them.
Answer Type	Text solution:1
Upvotes	150

Show that the line r=(2i^+5j^​+7k^)+λ(i^+3j^​+4k^) is parallel to the plane r⋅(i^+j^​−k^)=7. Also, find the distance between them.

Text solutionVerified

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Show that the line $r = (2 \hat{i} + 5 \hat{j} + 7 \hat{k}) + λ (\hat{i} + 3 \hat{j} + 4 \hat{k})$ is parallel to the plane $r \cdot (\hat{i} + \hat{j} - \hat{k}) = 7$ . Also, find the distance between them.