Assertion: If position vector is given by `hatr=sinhati+costhatj-7thatk`, then magnitude of acceleration `|veca|=1`
Reason: The angles which the vector `vecA=A_(1)veci+A_(3)hatk` makes with the co-ordinate
asex are given by `cos alpha=(A_(1))/(A), cos beta=(A_(2))/(A)& cos gamma=(A_(3))/(A)`.

Options

A. If both Assertion `&` Reason are True `&` the Reason is a correct explanation of the Assertion.
B. If both Assertion `&` Reason are True but Reason is not a correct explanation of the Assertion.
C. If Assertion is True but the Reason is False.
D. If both Assertion `&` Reason are false.

Explanation

To solve the given problem, we need to analyze the assertion and the reason provided in the question step by step.

### Step 1: Identify the Position Vector

The position vector is given as:
\[ \vec{r} = \hat{i} \sin t + \hat{j} \cos t - 7 \hat{k} \]

### Step 2: Calculate the Velocity Vector

To find the velocity vector \(\vec{v}\), we differentiate the position vector \(\vec{r}\) with respect to time \(t\):
\[ \vec{v} = \frac{d\vec{r}}{dt} = \frac{d}{dt}(\hat{i} \sin t + \hat{j} \cos t - 7 \hat{k}) \]
Calculating the derivatives:
\[ \vec{v} = \hat{i} \cos t - \hat{j} \sin t + 0 \hat{k} = \hat{i} \cos t - \hat{j} \sin t \]

### Step 3: Calculate the Acceleration Vector

Next, we find the acceleration vector \(\vec{a}\) by differentiating the velocity vector \(\vec{v}\) with respect to time \(t\):
\[ \vec{a} = \frac{d\vec{v}}{dt} = \frac{d}{dt}(\hat{i} \cos t - \hat{j} \sin t) \]
Calculating the derivatives:
\[ \vec{a} = -\hat{i} \sin t - \hat{j} \cos t \]

### Step 4: Calculate the Magnitude of the Acceleration Vector

Now, we calculate the magnitude of the acceleration vector \(|\vec{a}|\):
\[ |\vec{a}| = \sqrt{(-\sin t)^2 + (-\cos t)^2} = \sqrt{\sin^2 t + \cos^2 t} \]
Using the Pythagorean identity \(\sin^2 t + \cos^2 t = 1\):
\[ |\vec{a}| = \sqrt{1} = 1 \]

### Step 5: Analyze the Assertion

The assertion states that the magnitude of acceleration \(|\vec{a}| = 1\). From our calculations, we found that this statement is true.

### Step 6: Analyze the Reason

The reason provided states that the angles which the vector \(\vec{A}\) makes with the coordinate axes are given by:
\[ \cos \alpha = \frac{A_1}{A}, \quad \cos \beta = \frac{A_2}{A}, \quad \cos \gamma = \frac{A_3}{A} \]
This is indeed a correct statement regarding direction cosines of a vector. However, it does not directly explain why the magnitude of acceleration is 1.

### Conclusion

Both the assertion and reason are true, but the reason does not correctly explain the assertion. Therefore, the correct conclusion is that the assertion is true and the reason is true but not a correct explanation of the assertion.

### Final Answer

The assertion is true, and the reason is true but does not explain the assertion.

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