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Chapter 3: Q3P (page 99)

The diagonals of a parallelogram bisect each other.

Short Answer

Expert verified

Parallelogram method of adding vectors aligns the graph as given below.

Hence, the given theorem is proved.

Step by step solution

01

Concept and formula used

Where Aand Bare the vectors that forms the parallelogram,d1andd2are its first and the second diagonals and the pointsm1andm2are the mid points of the first and the second diagonals respectively.

02

Usage of Vector in Geometry

We have

d1=A+Bd2=B-A

Now, let us call the vector from the origin to the point m1J1. And the vector from the origin to the point m2J2. :

J1=d12=A+B2

J2=B-d22=B-B-A2=A+B2

J1=J2=A+B2

This signifies that the midpoints of each diagonal are in the same place in relation to O.

Then the graph should be

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Most popular questions from this chapter

Find the Eigen values and Eigen vectors of the following matrices. Do some problems by hand to be sure you understand what the process means. Then check your results by computer.

(222-1)

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 .

(a) Prove that.(AB)t=BtAt Hint: See(9.10).

(b) Verify (9.11), that is, show that (9.10) applies to a product of any number of matrices. Hint: Use (9.10)and (9.8).

Find AB,BA,A+B,A-B,A2,B2,5-A,3-B. Observe thatAB≠BA.Show that(A-B)(A+B)≠(A+B)(A-B)≠A2-B2. Show that det(AB)=det(BA)=(detA)(detB), but that det(A+B)≠detA+detB. Show that det(5A)≠5detA and find n so that localid="1658983435079" det(5A)=5ndetA.Find similar results for det(3B). Remember that the point of doing these simple problems by hand is to learn how to manipulate determinants and matrices correctly. Check your answers by computer.

localid="1658983077106" A=(25-13),B=(-1402)

Evaluate the determinants in Problems 1 to 6 by the methods shown in Example 4. Remember that the reason for doing this is not just to get the answer (your computer can give you that) but to learn how to manipulate determinants correctly. Check your answers by computer.

Answer

Step-by-Step Solution

Step 2: Find the determinant.

The objective is to determine the determinant of .

Add two times the third column in the second column, to get

Now, do the Laplace development using the second column to get

Hence, the value of the determinant is .
See all solutions

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