In each case, find the x- and y-components of vector A→: (a) A→=5.0i^-6.3j^; (b) A→=11.2j^-9.91i^; (c) A→=-15.0i^+22.4j^; (d) A→=5.0B→, where B→=4i^-6j^.

Answer The component of the given vector in the x-direction is Ax = 5 and the component in the y-direction is Ay = -6.3 .The component of the given vector in the x-direction is Ax = -9.11 and the component in the y-direction is Ay = 11.2 .The co...

Q36E - University Physics with Modern Physics

Q36E

Question

In each case, find the x- and y-components of vector $\vec{A}$ : (a) $\vec{A} = 5.0 \hat{i} - 6.3 \hat{j}$ ; (b) $\vec{A} = 11.2 \hat{j} - 9.91 \hat{i}$ ; (c) $\vec{A} = - 15.0 \hat{i} + 22.4 \hat{j}$ ; (d) $\vec{A} = 5.0 \vec{B}$ , where $\vec{B} = 4 \hat{i} - 6 \hat{j}$ .

Step-by-Step Solution

Verified

Answer

Answer

The component of the given vector in the x-direction is A_x = 5 and the component in the y-direction is A_y= -6.3 .
The component of the given vector in the x-direction is A_x = -9.11 and the component in the y-direction is A_y= 11.2 .
The component of the given vector in the x-direction is A_x = -15.0 and the component in the y-direction is A_y=22.4
The component of the given vector in the x-direction is A_x = 20.0 and the component in the y-direction is A_y= -30.0

1Step-by-Step Solutions Step 1 Identification of given data

$• T h e g i v e n v e c t o r s a r e, \vec{A} = 5.0 \hat{i} - 6.3 \hat{j}, \vec{A} = 11.2 \hat{j} - 9.91 \hat{i}, \vec{A} = - 15.0 \hat{i} + 22.4 \hat{j} a n d \vec{B} = 4 \hat{i} - 6 \hat{j}$

$• T h e r e l a t i o n b e t w e e n v e c t o r \vec{A} a n d v e c t o r \vec{B} i s g i v e n b y \vec{A} = 5.0 \vec{B}$

2Step 2: Vector and vector component

For a given vector $\vec{A} = A_{x} \hat{i} + A_{y} \hat{j}$ , A_x is the component in x- direction and A_y is the component in y-direction and are the unit vectors in x and y directions respectively

3Step 3: Determination of vector components of given vector

Part(a)

The given vector $\overset{⇀}{A}$ can be represented as,

$\vec{A} = 5.0 \hat{i} - 6.3 \hat{j}$

The given vector can be represented in the form of $\vec{A} = A_{x} \hat{i} + A_{y} \hat{j}$ as,

$\vec{A} = 5.0 \hat{i} + (- 6.3) \hat{j}$

The components of the given vector A_x and A_y will be,

A_x = 5.0

A_y = -6.3

Thus, the component of the given vector in the x-direction is A_x = 5.0 and the component in the y-direction is A_y = -6.3 .

4Step 4: Determination of vector components of given vector

Part(b)

The given vector $\vec{A}$ can be represented as,

$\vec{A} = 11.2 \hat{j} - 9.91 \hat{i}$

The given vector can be represented in the form of $\vec{A} = A_{x} \hat{i} + A_{y} \hat{j}$ as,

$\vec{A} = (- 9.91) \hat{i} + 11.2 \hat{j}$

The components of the given vector A_x and A_y can be represented as,

$\begin{array}{l} A_{x} = - 9.11 \\ A_{y} = 11.2 \end{array}$

Thus, the component of the given vector in the x-direction is A_X = -9.11 and the component in the y-direction is A_y = 11.2

5Step 5: Determination of vector components of given vector

Part(c)

The given vector $\vec{A}$ can be represented as,

$\vec{A} = - 15.0 \hat{i} + 22.4 \hat{j}$

The given vector can be represented in the form of $\vec{A} = A_{x} \hat{i} + A_{y} \hat{j}$ as,

$\vec{A} = (- 15.0) \hat{i} + 22.4 \hat{j}$

The components of the given vector A_x and A_y can be represented as,

$\begin{array}{l} A_{x} = - 15.0 \\ A_{y} = 22.4 \end{array}$

Thus, the component of the given vector in the x-direction is A_x = -15.0 and the component in the y-direction is A_y = 22.4 .

6Step 6: Determination of vector components of given vector

Part(d)

The given vector $\vec{B}$ can be represented as,

$\vec{B} = 4 \hat{i} - 6 \hat{j}$

The relation between vector $\vec{A}$ and vector $\vec{B}$ can be expressed as,

$\vec{A} = 5.0 \vec{B}$

$S u b s t i t u t e 4 \hat{i} - 6 \hat{j} f o r \vec{B}$

$\begin{matrix} \vec{A} = 5.0 \vec{B} \\ = 5.0 (4 \hat{i} - 6 \hat{j}) \\ = (5.0 \times 4) \hat{i} - (5.0 \times 6) \hat{j} \\ = 20.0 \hat{i} - 30.0 \hat{j} \end{matrix}$