The line `vecr=alpha(hati+hatj+hatk)+3hatk and vecr=beta(hati-2hatj+hatk)+3hatk` (A) intersect at rilghat angles (B) are skew (C) are parallel (D) none of these

Options

A. Intersects at rifht angle
B. Are skew
C. Are parallel
D. Are not perpendicular

Answer

To determine the relationship between the two lines given by the equations:

1. \(\vec{r} = \alpha(\hat{i} + \hat{j} + \hat{k}) + 3\hat{k}\)
2. \(\vec{r} = \beta(\hat{i} - 2\hat{j} + \hat{k}) + 3\hat{k}\)

we will follow these steps:

### Step 1: Identify the direction vectors and points of intersection

From the equations, we can identify the direction vectors and points through which the lines pass.

- For line 1:
- Direction vector, \(\vec{y_1} = \hat{i} + \hat{j} + \hat{k}\)
- Point on the line, \(\vec{p_1} = 3\hat{k}\)

- For line 2:
- Direction vector, \(\vec{y_2} = \hat{i} - 2\hat{j} + \hat{k}\)
- Point on the line, \(\vec{p_2} = 3\hat{k}\)

### Step 2: Check if the lines intersect

Since both lines pass through the point \(3\hat{k}\), they do intersect.

### Step 3: Calculate the angle between the two direction vectors

To find the angle between the two lines, we can use the dot product formula:

\[ \cos \theta = \frac{\vec{y_1} \cdot \vec{y_2}}{|\vec{y_1}| |\vec{y_2}|} \]

Calculating the dot product:

\[ \vec{y_1} \cdot \vec{y_2} = (\hat{i} + \hat{j} + \hat{k}) \cdot (\hat{i} - 2\hat{j} + \hat{k}) \]
\[ = 1 \cdot 1 + 1 \cdot (-2) + 1 \cdot 1 = 1 - 2 + 1 = 0 \]

### Step 4: Calculate the magnitudes of the direction vectors

Now we calculate the magnitudes of \(\vec{y_1}\) and \(\vec{y_2}\):

\[ |\vec{y_1}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3} \]
\[ |\vec{y_2}| = \sqrt{1^2 + (-2)^2 + 1^2} = \sqrt{1 + 4 + 1} = \sqrt{6} \]

### Step 5: Substitute into the cosine formula

Now substituting back into the cosine formula:

\[ \cos \theta = \frac{0}{\sqrt{3} \cdot \sqrt{6}} = 0 \]

### Step 6: Determine the angle

Since \(\cos \theta = 0\), it implies that:

\[ \theta = \frac{\pi}{2} \text{ or } 90^\circ \]

### Conclusion

The two lines intersect at right angles.

Thus, the correct option is (A) intersect at right angles.

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