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Chapter 9: Problem 46

Find an equation of the plane. The plane contains the \(y\) -axis and makes an angle of \(\pi / 6\) with the positive \(x\) -axis.

Short Answer

Expert verified
The equation of the plane is \(x = 0\).

Step by step solution

01

Understanding the Problem

We notice from the problem that the plane contains the \(y\)-axis and makes an angle of \(\pi / 6\) with the positive \(x\) -axis.
02

Find Normal Vector

Since the plane contains the \(y\)-axis, the normal to the plane (vector perpendicular to the plane) will be in the form of \(N = a\hat{i} + c\hat{k}\), where \(a\) and \(c\) are the coefficients along \(x\) and \(z\)-axes.
03

Use Angle Definition

The direction cosines of the normal vector to the plane are given by the ratios of the coefficients of the vector to its magnitude. As the plane makes an angle of \(\pi / 6\) with the \(x\)-axis, the cosine of this angle equals \(a/||N||\), where \(||N||\) is the magnitude of the normal vector.
04

Solve for Vector Coefficients

Assuming unit vector, \(||N|| = 1\), hence \(a = \cos (\pi / 6) = \sqrt{3}/2\). The coefficient along the \(z\)-axis \(c\) can be any real number as no information regarding the \(z\)-axis is given in the problem. Hence, we choose \(c=0\) for simplicity.
05

Write Down Equation of Plane

The general equation of a plane given a normal vector \(N=a\hat{i}+b\hat{j}+c\hat{k}\) and a point \(P=(x_0,y_0,z_0)\) on the plane is \((a(x-x_0)+b(y-y_0)+c(z-z_0)=0\). Since \(N=\sqrt{3}/2\hat{i}+0\hat{j}+0\hat{k}\) and knowing that our plane passes through the \(y\)-axis (for simplicity, we take point on \(y\)-axis as \(P=(0,0,0)\)), we substitute these values into the equation of the plane to get \(\sqrt{3}/2*x=0\)

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Most popular questions from this chapter

In Exercises \(1-6,\) find \((\mathbf{a}) \mathbf{u} \cdot \mathbf{v},(\mathbf{b}) \mathbf{u} \cdot \mathbf{u},(\mathbf{c})\|\mathbf{u}\|^{2},(\mathbf{d})(\mathbf{u} \cdot \mathbf{v}) \mathbf{v}\) and \((e) u \cdot(2 v)\). $$ \mathbf{u}=\langle 5,-1\rangle, \quad \mathbf{v}=\langle-3,2\rangle $$

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