Question

suppose that each component of a certain vector is doubled. (a) by what multiplication factor does the magnitude of the vector change. ( b) by what multiplicative factor does the direction of the vector change

Answer 1

Answer:

a) The magnitude of the vector is changed my a multiplication factor of 2

b) The direction of the vector is changed by a multiplication factor of 1

Step-by-step explanation:

Let the first component of a vector be x

Let the second component of the vector be y

The vector, A = xi + yj

The magnitude of the vector A will be:

`|A|\text{ = }\sqrt[]{x^2+y^2}`

The direction of the vector A will be:

`\theta_A\text{ = }\tan ^(-1)(y)/(x)`

If each component of the vector is doubled

Double of the first component = 2x

Double of the second component = 2y

The new vector, B = 2xi + 2yj

The magnitude of the new vector B will be:

`\begin{gathered} |B|\text{ = }\sqrt[]{(2x)^2+(2y)^2} \\ |B|\text{ = }\sqrt[]{4x^2+4y^2} \\ |B|\text{ = }\sqrt[]{4(x^2+y^2)} \\ |B|\text{ = 2}\sqrt[]{x^2+y^2} \end{gathered}`

The direction of the new vector B will be:

`\begin{gathered} \theta_B=\text{ }\tan ^(-1)(2y)/(2x) \\ \theta_B=\text{ }\tan ^(-1)(y)/(x) \end{gathered}`

Since:

`|A|\text{ = }\sqrt[]{x^2+y^2}\text{ and }|B|=2\sqrt[]{x^2+y^2}`

then:

|B| = 2|A|

The maginitude of the vector is double. That is, it is changed by a multiplication factor of 2

Since:

`\theta_A=\text{ }\tan ^(-1)(y)/(x)\text{ and }\theta_B=\text{ }\tan ^(-1)(y)/(x)`

then:

The direction of the vector does not change. That is, the multiplicative factor is 1