Q37E

Question

Write each vector in Fig. E1.24 in terms of the unit vectors $\hat{i}$ and $\hat{j}$

Step-by-Step Solution

Verified

Answer

Answer

The representation of $\vec{A}$ in terms of unit vectors is $\vec{A} = 0 \hat{i} - (8.0 m) \hat{j}$ .
The representation of vector $\vec{B}$ in terms of unit vectors is $\vec{B} = (7.50 m) \hat{i} + (13.0 m) \hat{j}$ .
The representation of vector $C$ in terms of unit vectors is $\vec{C} = (- 10.9 m) \hat{i} + (- 5.07 m) \hat{j}$
The representation of vector $\vec{D}$ in terms of unit vectors is, $\vec{D} = (- 7.99 m) \hat{i} + (6.02 m) \hat{j}$ .

1Step-by-Step Solution Step 1: Identification of given data

Length of vector $\vec{A}$ is 8.00 m.
Length of vector $\vec{B}$ is 15.0 m.
Length of vector $\vec{C}$ is 12.0 m.
Length of vector $\vec{D}$ is 10.0 m.

2Step 2: Representation of vector A ⇀ in terms of unit vectors

The representation of $\vec{A}$ in terms of unit vectors is,

$\vec{A} = A_{x} \hat{i} + A_{y} \hat{j}$ .

From the given diagram the components,

$\begin{array}{l} A_{x} = 0 \\ A_{y} = - 8.0 m \end{array}$

Thus, the representation of $\vec{A}$ in terms of unit vectors is $\vec{A} = 0 \hat{i} - (8.0 m) \hat{j}$ .

3Step 3: Representation of vector B → in terms of unit vectors

The representation of $\vec{B}$ in terms of unit vectors is,

$\vec{B} = B_{x} \hat{i} + B_{y} \hat{j}$ .

From the given diagram angle between $\vec{B}$ and x-axis is,

$\begin{matrix} θ = 90 ° - 30 ° \\ = 60 ° \end{matrix}$

Thus, the components of vector $\vec{B}$ is,

$\begin{array}{l} B_{x} = B \cos θ \\ B_{y} = B \sin θ \end{array}$

Substitute 15.0 m for B, and 60^o for $θ$ in the above equations,

$\begin{matrix} B_{x} = (15.0 m) \cos (60 °) \\ = 7.50 m \\ B_{y} = (15.0 m) \sin (60 °) \\ = 13.0 m \end{matrix}$

Thus, the representation of vector $\vec{B}$ in terms of unit vectors is, $\vec{B} = (7.50 m) \hat{i} + (13.0 m) \hat{j}$ .

4Step 4: Representation of vector C → in terms of unit vectors

The representation of $\vec{C}$ in terms of unit vectors is,

$\vec{C} = C_{x} \hat{i} + C_{y} \hat{j}$ .

From the given diagram angle between $\vec{C}$ and x-axis is,

$θ = 25^{o}$

Thus, the components of vector $\vec{C}$ is,

$\begin{array}{l} C_{x} = C \cos θ \\ C_{y} = C \sin θ \end{array}$

Substitute -12.0 m for C and 25^o for $θ$ in the above equations,

$\begin{matrix} C_{x} = (- 12.0 m) \cos (25 °) \\ = - 10.9 m \\ C_{y} = (- 12.0 m) \sin (25 °) \\ = - 5.07 m \end{matrix}$

Thus, the representation of vector $\vec{C}$ in terms of unit vectors is, $\vec{C} = (- 10.9 m) \hat{i} + (- 5.07 m) \hat{j}$ .

5Step 5: Representation of vector D → in terms of unit vectors

The representation of $\vec{D}$ in terms of unit vectors is,

$\vec{D} = D_{x} \hat{i} + D_{y} \hat{j}$

From the given diagram angle between $\vec{D}$ and x-axis is,

$\begin{matrix} θ = 90 ° - 53 ° \\ = 37 ° \end{matrix}$

Thus, the components of vector $\vec{D}$ is,

$\begin{array}{l} D_{x} = D \cos θ \\ D_{y} = D \sin θ \end{array}$

Substitute 10.0m for D and 37^o for $θ$ in the above equations,

$\begin{matrix} D_{x} = (10.0 m) \cos (37 °) \\ = - 7.99 m \\ D_{y} = (- 12.0 m) \sin (37 °) \\ = 6.02 m \end{matrix}$

Thus, the representation of vector $\vec{D}$ in terms of unit vectors is, data-custom-editor="chemistry" $\vec{D} = (- 7.99 m) \hat{i} + (6.02 m) \hat{j}$ .

Q 35E

Q38E

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