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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, $d_{1}$ and $d_{2}$ are its first and the second diagonals and the points $m_{1}$ and $m_{2}$ are the mid points of the first and the second diagonals respectively.

02

Usage of Vector in Geometry

We have

$d_{1} = A + B d_{2} = B - A$

Now, let us call the vector from the origin to the point $m_{1} J_{1}$ . And the vector from the origin to the point $m_{2} J_{2}$ . :

$J_{1} = \frac{d_{1}}{2} = \frac{A + B}{2}$

$J_{2} = B - \frac{d_{2}}{2} = B - \frac{B - A}{2} = \frac{A + B}{2}$

$J_{1} = J_{2} = \frac{A + B}{2}$

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

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.

$\frac{1}{\sqrt{2}} (\begin{matrix} - 1 & - 1 \\ 1 & - 1 \end{matrix})$

Draw diagrams and prove (4.1).

A particle is traveling along the line (x-3)/2=(y+1)/(-2)=z-1. Write the equation of its path in the form $r = r_{0} + At$ . Find the distance of closest approach of the particle to the origin (that is, the distance from the origin to the line). If t represents time, show that the time of closest approach is $t = - (r_{0} \times A) / {| A |}^{2}$ . Use this value to check your answer for the distance of closest approach. Hint: See Figure 5.3. If P is the point of closest approach, what is $A \times r^{2}$ ?

Find a vector perpendicularto both i+j and i-2k .

The Caley-Hamilton theorem states that "A matrix satisfies its own characteristic equation." Verify this theorem for the matrix $M$ in equation (11.1). Hint: Substitute the matrix $M$ forrole="math" localid="1658822242352" $λ$ in the characteristic equation (11.4) and verify that you have a correct matrix equation. Further hint: Don't do all the arithmetic. Use (11.36) to write the left side of your equation as $C (D^{2} - 7 D + 6) C^{- 1}$ and show that the parenthesis $= 0$ . Remember that, by definition, the eigenvalues satisfy the characteristic equation.

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