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Chapter 3: Q32P (page 132)

For each of the following matrices, find its determinant to see whether it produces a rotation or reflection. If a rotation, find the axis and the angle of rotation. If a reflection, find the reflecting plane and the rotation (if any) about the normal to this plane.

(00-10-10-100)

Short Answer

Expert verified

The given matrix produces a rotation along the axis i-K and the angle of rotation is 180°.

Step by step solution

01

 Step 1: Matrix Transformations

Matrix transformation can be of two types rotation and reflection only for square matrices. When the determinant value of the matrix is1 then it is termedrotationand if the value is-1then it isreflection.

02

Given Parameter

The given matrix is M=00-10-10-100.

03

Calculate detM.

detM=0-1×0-0×0-00×0-0×-1-10×0--1×-1=0-0-1-1=1

Hence M is a rotation.

To determine the axis of rotation, use the formula, Mr =r, where r=xyz.

Resulted in the matrix equation is00-10-10-100xyz=xyz

The equations obtained are -z=x,-y=y, and-x=z.

It is seen that the vector (1,0,-1) remains unchanged after the transformation hence, the axis of rotation is i-k.

Calculate.

M2=00-10-10-10000-10-10-100=100010001

Hence the angle of rotation is 180°.

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Most popular questions from this chapter

(a) If Cis orthogonal and Mis symmetric, show that C-1MCis symmetric.

(b) IfC is orthogonal and Mantisymmetric, show thatC-1MCis antisymmetric.

Find the symmetric equations (5.6) or (5.7) and the parametric equations (5.8) of a line, and/or the equation (5.10) of the plane satisfying the following given conditions.

Line through and parallel to the line .

Answer

The symmetric equations of the line is .

The parametric equation is .

Step-by-Step Solution

Step 1: Concept of the symmetric and parametric equations

The symmetric equations of the line passing through and parallel to is

The parametric equations of the line are

Step 2: Determine the symmetric equation of a straight line

The given point is and the line is .

The given line is in the form of . So, we get

The symmetric equations of the straight line passing through and parallel to is given by

Thus, the required solution is .

Step 3: Determine the parametric equation of a straight line.

The parametric equations of the straight line passing through and parallel to is given by

Or

.

Thus, the required solution is .

Show that a real Hermitian matrix is symmetric. Show that a real unitary matrix is orthogonal. Note: Thus, we see that Hermitian is the complex analogue of symmetric, and unitary is the complex analogue of orthogonal. (See Section 11.)


In Problems8to15,use(8.5) show that the given functions are linearly independent.

14.eix,e−ix

In Problems 8to15,use (8.15) to show that the given functions are linearly independent.

sinx,cosx,xsinx,xcosx.
See all solutions

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