A vector `vecr` is inclined to x-axis at `45^(@)` and y-axis at `60^(@)` if `|vecr| =8` units. Find `vecr.`

Answer

To find the vector \(\vec{r}\) that is inclined to the x-axis at \(45^\circ\) and to the y-axis at \(60^\circ\) with a magnitude of \(8\) units, we can follow these steps:

### Step 1: Determine Direction Cosines
The direction cosines \(L\), \(M\), and \(N\) are defined as follows:
- \(L = \cos(45^\circ)\)
- \(M = \cos(60^\circ)\)
- \(N = \cos(\alpha)\) (where \(\alpha\) is the angle with the z-axis)

Calculating the values:
- \(L = \cos(45^\circ) = \frac{1}{\sqrt{2}}\)
- \(M = \cos(60^\circ) = \frac{1}{2}\)

### Step 2: Use the Property of Direction Cosines
The sum of the squares of the direction cosines is equal to 1:
\[ L^2 + M^2 + N^2 = 1 \]
Substituting the known values:
\[ \left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{2}\right)^2 + N^2 = 1 \]
Calculating the squares:
\[ \frac{1}{2} + \frac{1}{4} + N^2 = 1 \]
Finding a common denominator (which is 4):
\[ \frac{2}{4} + \frac{1}{4} + N^2 = 1 \]
This simplifies to:
\[ \frac{3}{4} + N^2 = 1 \]
Thus, we can find \(N^2\):
\[ N^2 = 1 - \frac{3}{4} = \frac{1}{4} \]
Taking the square root:
\[ N = \frac{1}{2} \]

### Step 3: Write the Vector in Component Form
The vector \(\vec{r}\) can be expressed in terms of its direction cosines:
\[ \vec{r} = L \hat{i} + M \hat{j} + N \hat{k} \]
Substituting the values of \(L\), \(M\), and \(N\):
\[ \vec{r} = \frac{1}{\sqrt{2}} \hat{i} + \frac{1}{2} \hat{j} + \frac{1}{2} \hat{k} \]

### Step 4: Multiply by the Magnitude
Given that the magnitude of \(\vec{r}\) is \(8\) units, we multiply the vector by \(8\):
\[ \vec{r} = 8 \left(\frac{1}{\sqrt{2}} \hat{i} + \frac{1}{2} \hat{j} + \frac{1}{2} \hat{k}\right) \]
Calculating each component:
\[ \vec{r} = 8 \cdot \frac{1}{\sqrt{2}} \hat{i} + 8 \cdot \frac{1}{2} \hat{j} + 8 \cdot \frac{1}{2} \hat{k} \]
This simplifies to:
\[ \vec{r} = 4\sqrt{2} \hat{i} + 4 \hat{j} + 4 \hat{k} \]

### Final Result
Thus, the vector \(\vec{r}\) is:
\[ \vec{r} = 4\sqrt{2} \hat{i} + 4 \hat{j} + 4 \hat{k} \]

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