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Chapter 3: Q1P (page 99)

Draw diagrams and prove (4.1).

Short Answer

Expert verified

The diagram is verified using (4.1) theorem.

Step by step solution

01

Formula for the magnitude of the components of the vector:

If $A_{x}, A_{y},$ and $A_{z}$ are the magnitudes of components of vector $\vec{A}$ along the coordinate axis, then $\vec{A} = A_{x} \hat{i} + A_{y} \hat{j} + A_{z} \hat{k}$ , where $\hat{i}, \hat{j}, \hat{k}$ respectively are the unit vectors along $x, y$ and zaxis.

02

The diagram of vector A→ in two dimensions:

Using Pythagoras theorem, the length of A , named $A = | A |$ is given by,

$A^{2} = A_{x}^{2} + A_{y}^{2} A = \sqrt{A_{x}^{2} + A_{y}^{2}}$

Where the dotted lines represent the components of the vector $A_{x}$ and $A_{y}$

03

The diagram of vector A→ in three dimensions:

Using Pythagoras theorem in $3 D$ , the length of A , named $A = | A |$ is given by,

$A^{2} = A_{x}^{2} + A_{y}^{2} + A_{z}^{2} A = \sqrt{A_{x}^{2} + A_{y}^{2} + A_{z}^{2}}$

Where the dotted lines represent the components of the vector $A_{x}, A_{y}$ and $A_{z}$ .

Hence, the condition is proved.

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Most popular questions from this chapter

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

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

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

$(\begin{matrix} 13 & 4 & - 2 \\ 4 & 13 & - 2 \\ - 2 & - 2 & 10 \end{matrix})$

(a): As in problem 12,
$D^{2} + 2 D + 1$ linear?

(b): Is $x^{2} D^{2} - 2 x D + 7$ a linear operator?

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

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

For the Pauli spin matrix Ain Problem 6 , find the matricessin(kA) ,cos(kA) , $e^{kA}, e^{ikA},$ where $= \sqrt{- 1}$ .

See all solutions

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