What is the angle between the vector `vec( r)=(4hat(i) +8hat(j) +hat(k)) ` and the x-axis ?

Explanation

To find the angle between the vector \(\vec{r} = 4\hat{i} + 8\hat{j} + \hat{k}\) and the x-axis, we can follow these steps:

### Step 1: Identify the components of the vector
The vector \(\vec{r}\) has components:
- \(a = 4\) (the coefficient of \(\hat{i}\))
- \(b = 8\) (the coefficient of \(\hat{j}\))
- \(c = 1\) (the coefficient of \(\hat{k}\))

### Step 2: Calculate the magnitude of the vector
The magnitude of the vector \(\vec{r}\) can be calculated using the formula:
\[ |\vec{r}| = \sqrt{a^2 + b^2 + c^2} \]
Substituting the values:
\[ |\vec{r}| = \sqrt{4^2 + 8^2 + 1^2} = \sqrt{16 + 64 + 1} = \sqrt{81} = 9 \]

### Step 3: Find the cosine of the angle with the x-axis
The cosine of the angle \(\alpha\) with the x-axis is given by:
\[ \cos \alpha = \frac{a}{|\vec{r}|} \]
Substituting the values we found:
\[ \cos \alpha = \frac{4}{9} \]

### Step 4: Calculate the angle \(\alpha\)
To find the angle \(\alpha\), we take the inverse cosine:
\[ \alpha = \cos^{-1}\left(\frac{4}{9}\right) \]

### Final Answer
Thus, the angle between the vector \(\vec{r}\) and the x-axis is:
\[ \alpha = \cos^{-1}\left(\frac{4}{9}\right) \]

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