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Chapter 3: Q5P (page 99)

In a parallelogram, the two lines from one corner to the midpoints of the two opposite sides trisect the diagonal they cross.

Short Answer

Expert verified

We use the rules of vector addition, and properties of triangles to prove the given theorem.

Step by step solution

01

Concept used

The parallelogram we are considering is

02

Usage of vectors in Geometry

Consider the parallelogram given below.

From thismAandmBare the midpoints of the vectors A and B respectively.

Now from the figure,

EmAOF=x2EFx212AA=x2EFx2Fx2=2脳x2E

d=Fx2+x2E=2脳x2E+x2E=3脳x2

x2E=d3

Also,

FmBOE=Fx1x1E12BB=Fx1x1Ex1E=2脳Fx1

d=Fx1+x1E=Fx1+2脳Fx1=3脳Fx1Fx1=d3

And sincex2E=Fx1=d3then the two lines from one corner to the midpoints of thetwo opposite sides trisect the diagonal as they cross.

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Most popular questions from this chapter

Show that an orthogonal matrix M with all real eigenvalues is symmetric. Hints: Method 1. When the eigenvalues are real, so are the eigenvectors, and the unitary matrix which diagonalizes M is orthogonal. Use (11.27). Method 2. From Problem 46, note that the only real eigenvalues of an orthogonal M are 卤1. Thus show that M=M-1 . Remember that M is orthogonal to show that M=MT.

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 .

Repeat the last part of Problem for the matrix M=(3-1-13)

Let each of the following matrices represent an active transformation of vectors in (x,y)plane (axes fixed, vector rotated or reflected).As in Example 3, show that each matrix is orthogonal, find its determinant and find its rotation angle, or find the line of reflection.

12(-1-11-1)

The Pauli spin matrices in quantum mechanics areA=(1001) ,B=(0-ii0) ,C=(100-1) .For the Pauli spin matrix C , find the matricessinkC ,coskC ,ekC, andeikC . Hint: Show that if a matrix is diagonal, sayD=(a00b), then f(D)=(f(a)00fb).
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