If a body placed at the origin is acted upon by a force `vecF=(hati+hatj+sqrt2hatk)`, then which of the following statements are correct?
Magnitude of `vecF` is (2+ `sqrt2`)
Magnitude of `vecF` is 2.
`vecF` makes an angle of `45^@` with the Z-axis
`vecF` makes an angle of `30^@` with the Z-axis.
Select the correct answer using the codes given below

Options

A. 1 and 3
B. 2 and 3
C. 1 and 4
D. 2 and 4

Answer

To solve the problem, we need to analyze the force vector given and determine the magnitude of the force and the angle it makes with the Z-axis.

**Step 1: Identify the force vector.**
The force vector is given as:
\[ \vec{F} = \hat{i} + \hat{j} + \sqrt{2}\hat{k} \]

**Step 2: Calculate the magnitude of the force vector.**
The magnitude of a vector \(\vec{F} = a\hat{i} + b\hat{j} + c\hat{k}\) is calculated using the formula:
\[ |\vec{F}| = \sqrt{a^2 + b^2 + c^2} \]
For our vector:
- \(a = 1\) (coefficient of \(\hat{i}\))
- \(b = 1\) (coefficient of \(\hat{j}\))
- \(c = \sqrt{2}\) (coefficient of \(\hat{k}\))

Now substituting these values into the formula:
\[ |\vec{F}| = \sqrt{1^2 + 1^2 + (\sqrt{2})^2} = \sqrt{1 + 1 + 2} = \sqrt{4} = 2 \]

**Step 3: Determine the angle with the Z-axis.**
To find the angle \(\theta\) that the force vector makes with the Z-axis, we use the dot product formula:
\[ \vec{F} \cdot \hat{k} = |\vec{F}| |\hat{k}| \cos \theta \]
Here, \(\hat{k}\) is the unit vector in the Z-direction, which has a magnitude of 1.

Calculating the dot product \(\vec{F} \cdot \hat{k}\):
\[ \vec{F} \cdot \hat{k} = 0\hat{i} + 0\hat{j} + \sqrt{2}\hat{k} = \sqrt{2} \]

Now substituting into the dot product equation:
\[ \sqrt{2} = 2 \cdot 1 \cdot \cos \theta \]
This simplifies to:
\[ \sqrt{2} = 2 \cos \theta \]
Dividing both sides by 2:
\[ \cos \theta = \frac{\sqrt{2}}{2} \]
From trigonometry, we know that:
\[ \theta = 45^\circ \]

**Final Results:**
1. The magnitude of \(\vec{F}\) is \(2\).
2. The angle \(\theta\) that \(\vec{F}\) makes with the Z-axis is \(45^\circ\).

**Conclusion:**
- The correct statements are:
- Magnitude of \(\vec{F}\) is \(2\).
- \(\vec{F}\) makes an angle of \(45^\circ\) with the Z-axis.

Thus, the correct answer codes are 2 and 3.

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