To solve the given problem, we need to evaluate the assertion (A) and reason (R) provided. Let's break it down step by step.
### Step 1: Find the Direction Cosines of the Vector
Given the vector \( \vec{A} = 2\hat{i} + 4\hat{j} - 5\hat{k} \).
The direction cosines of a vector are given by the formulas:
\[
l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \quad m = \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \quad n = \frac{c}{\sqrt{a^2 + b^2 + c^2}}
\]
where \( a, b, c \) are the components of the vector along the \( \hat{i}, \hat{j}, \hat{k} \) directions respectively.
### Step 2: Calculate the Magnitude of the Vector
First, we need to find the magnitude of the vector \( \vec{A} \):
\[
|\vec{A}| = \sqrt{a^2 + b^2 + c^2} = \sqrt{2^2 + 4^2 + (-5)^2}
\]
Calculating each term:
\[
2^2 = 4, \quad 4^2 = 16, \quad (-5)^2 = 25
\]
Adding these values:
\[
|\vec{A}| = \sqrt{4 + 16 + 25} = \sqrt{45}
\]
### Step 3: Calculate the Direction Cosines
Now we can find the direction cosines:
\[
l = \frac{2}{\sqrt{45}}, \quad m = \frac{4}{\sqrt{45}}, \quad n = \frac{-5}{\sqrt{45}}
\]
Thus, the direction cosines of the vector \( \vec{A} \) are:
\[
\left( \frac{2}{\sqrt{45}}, \frac{4}{\sqrt{45}}, -\frac{5}{\sqrt{45}} \right)
\]
### Step 4: Verify Assertion (A)
The assertion states that the direction cosines of the vector \( \vec{A} \) are:
\[
\left( \frac{2}{\sqrt{45}}, \frac{4}{\sqrt{45}}, -\frac{5}{\sqrt{45}} \right)
\]
Since we calculated the direction cosines and they match the assertion, we conclude that assertion (A) is true.
### Step 5: Evaluate Reason (R)
The reason states that a vector having zero magnitude and arbitrary direction is called a 'zero vector' or 'null vector'.
To verify this, we note that the magnitude of a vector \( \vec{A} \) is zero if all its components are zero:
\[
|\vec{A}| = \sqrt{a^2 + b^2 + c^2} = 0 \implies a = 0, b = 0, c = 0
\]
Thus, the reason (R) is also true.
### Conclusion
Both the assertion (A) and reason (R) are true. However, reason (R) does not explain assertion (A) since they are discussing different properties of vectors.
### Final Answer
- Assertion (A) is true.
- Reason (R) is true.
- Reason (R) is not the correct explanation of assertion (A).
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