A vector n of magnitude 8 units is inclined to the x-axis is 45o, y-axis at 60o and an acute angle with the z-axis. If a plane passes through a point (2,−1,1) and is normal to n, find its equation in vector form.

Question

A vector n of magnitude 8 units is inclined to the x-axis is 45o, y-axis at 60o and an acute angle with the z-axis. If a plane passes through a point (2​,−1,1) and is normal to n, find its equation in vector form.

Filo · Accepted Answer

If α,β, and γ are the angles made by the line segment with the coordinate axis.cosα,cosβ and cosγ are called as the direction cosines and are usually denoted by l,m and n.l=cosα,m=cosβ and n=cosγHere l=cos45o=2​1​, m=cos60o=21​ and let n=cosγ. Then,l2+m2+n2=1⇒21​+41​+cos2γ=1⇒cos2γ=41​⇒cosγ=21​We know that n^=∣n∣n​=(li^+mj^​+nk^).∴n=∣n∣(li^+mj^​+nk^)⇒n=8(2​1​i^+21​j^​+21​k^)=(42​i^+4j^​+4k^).So, the equation of the plane isr⋅n=a⋅n⇒r⋅(42​i^+4j^​+4k^)=(2​i^−j^​+k^)⋅(42​i^+4j^​+4k^)⇒r⋅(42​i^+4j^​+4k^)=(8−4+4)=8⇒r⋅(2​i^+j^​+k^)=2.

Question Text	A vector $n$ of magnitude $8$ units is inclined to the x-axis is $4 5^{o}$ , y-axis at $6 0^{o}$ and an acute angle with the z-axis. If a plane passes through a point $(2, - 1, 1)$ and is normal to $n$ , find its equation in vector form.
Answer Type	Text solution:1
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A vector $n$ of magnitude $8$ units is inclined to the x-axis is $4 5^{o}$ , y-axis at $6 0^{o}$ and an acute angle with the z-axis. If a plane passes through a point $(2, - 1, 1)$ and is normal to $n$ , find its equation in vector form.

Text solutionVerified

Practice questions from similar books

A vector n of magnitude 8 units is inclined to the x-axis is 45o, y-axis at 60o and an acute angle with the z-axis. If a plane passes through a point (2​,−1,1) and is normal to n, find its equation in vector form.

Text solutionVerified

Practice questions from similar books

A vector $n$ of magnitude $8$ units is inclined to the x-axis is $4 5^{o}$ , y-axis at $6 0^{o}$ and an acute angle with the z-axis. If a plane passes through a point $(2, - 1, 1)$ and is normal to $n$ , find its equation in vector form.