A vector of magnitude 14 lies in the xy-plane and makes an angle of `60^@` with x -axis . The components of the vector in the direction of x -axis and y-axis are

Options

A. `7, 7 sqrt3`
B. `7sqrt3 , 7`
C. `14sqrt3, 14//sqrt3`
D. `14//sqrt3 , 14sqrt3`

Answer

To find the components of a vector in the xy-plane, we can use trigonometric functions based on the angle the vector makes with the x-axis. Here’s the step-by-step solution:

### Step 1: Identify the given information
We have a vector with:
- Magnitude = 14
- Angle with the x-axis = 60 degrees

### Step 2: Use trigonometric functions to find components
The components of the vector can be found using the cosine and sine functions:
- The x-component (along the x-axis) can be calculated using the cosine of the angle:
\[ x = \text{Magnitude} \times \cos(\text{Angle}) = 14 \times \cos(60^\circ) \]
- The y-component (along the y-axis) can be calculated using the sine of the angle:
\[ y = \text{Magnitude} \times \sin(\text{Angle}) = 14 \times \sin(60^\circ) \]

### Step 3: Calculate the x-component
Using the value of \(\cos(60^\circ) = \frac{1}{2}\):
\[ x = 14 \times \frac{1}{2} = 7 \]

### Step 4: Calculate the y-component
Using the value of \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\):
\[ y = 14 \times \frac{\sqrt{3}}{2} = 7\sqrt{3} \]

### Step 5: State the final components
Thus, the components of the vector in the direction of the x-axis and y-axis are:
- \(x = 7\)
- \(y = 7\sqrt{3}\)

### Final Answer
The components of the vector are \(7\) (along the x-axis) and \(7\sqrt{3}\) (along the y-axis).

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