Griffiths problem 1.1 Using the definitions in Eqs. 1.1 and 1.4, and appropriate diagrams

Griffiths problem 1.1 Using the definitions in Eqs. 1.1 and 1.4, and appropriate diagrams


P
Asked by 2 years ago
945 points

Using the definitions in Eqs. 1.1 and 1.4, and appropriate diagrams, show that the dot product and cross product are distributive,

a) when the three vectors are coplanar;

b) in the general case

Griffiths problem
Pairoaj

1 Answer

P
Answered by 2 years ago
945 points

triangle diagram

In this problem use the following relationships

$$A \cdot B=AB \cos{\theta}$$

$$A \times B = AB \sin{\theta} \,\hat{n}$$

From the drawn diagram above,

$$\left\lvert B+C \right\rvert \,\cos{\theta_3}=\left\lvert B \right\rvert \cos{\theta_1}+\left\lvert C \right\rvert \cos{\theta_2}$$

$$\rightarrow \left\lvert A \right\rvert \left\lvert B+C \right\rvert \,\cos{\theta_3}=\left\lvert A \right\rvert \left\lvert B \right\rvert \cos{\theta_1}+\left\lvert A \right\rvert \left\lvert C \right\rvert \cos{\theta_2}$$

$$\rightarrow A\cdot(B+C)=A\cdot B+A \cdot C$$

hence the dot product is distributive. Similarly,

$$\left\lvert B+C \right\rvert \,\sin{\theta_3}=\left\lvert B \right\rvert \sin{\theta_1}+\left\lvert C \right\rvert \sin{\theta_2}$$

$$\rightarrow \left\lvert A \right\rvert \left\lvert B+C \right\rvert \,\sin{\theta_3}\,\hat{n}=\left\lvert A \right\rvert \left\lvert B \right\rvert \sin{\theta_1}\,\hat{n}+\left\lvert A \right\rvert \left\lvert C \right\rvert \sin{\theta_2}\,\hat{n}$$

$$\rightarrow A \times (B+C)=(A\times B) + (A\times C)$$

and hence the cross product is distributive

for the general case just calculate the value of

$$A \cdot (B+C)$$

where

$$A = A_x \,\hat{i}+A_y \,\hat{j} + A_z \,\hat{k}$$ $$B = B_x \,\hat{i}+B_y \,\hat{j} + B_z \,\hat{k}$$ $$C = C_x \,\hat{i}+C_y \,\hat{j} + C_z \,\hat{k}$$

for instance,

$$A \cdot (B+C) = (A_x,A_y,A_z) \cdot (B_x+C_x,B_y+C_y+B_z+C_z)$$

$$ = A_x(B_x+C_x)\,\hat{x}+A_y(B_y+C_y)\,\hat{y}+A_z(B_z+C_z)\,\hat{z}$$

$$ = (A_xB_x+A_xC_x)\,\hat{x}+(A_yB_y+A_yC_y)\,\hat{y}+(A_zB_z+A_zC_z)\,\hat{z}$$

now check that it is equal to $A \cdot B+A\cdot C$

the same can be done for the cross product $A \times (B+C) = (A\times B)+(A \times C)$

Your Answer

Surround your text in *italics* or **bold**, to write a math equation use, for example, $x^2+2x+1=0$ or $$\beta^2-1=0$$

Use LaTeX to type formulas and markdown to format text. See example.

Sign up or Log in

  • Answer the question above my logging into the following networks
Sign in
Sign in
Sign in

Post as a guest

  • Your email will not be shared or posted anywhere on our site
  •  

Stats
Views: 52
Asked: 2 years ago

Related