Let `veca=hati+hatj and vecb=2hati-hatk.` Then the point of intersection of the lines `vecrxxveca=vecbxxveca and vecrxxvecb=vecaxxvecb` is (A) `(3,-1,10` (B) `(3,1,-1)` (C) `(-3,1,1)` (D) `(-3,-1,-1)`

Options

A. (3,-1,1)
B. (3,1,-1)
C. (-3,1,1)
D. (-3,-1,-1)

Explanation

To find the point of intersection of the lines given by the equations \( \vec{r} \times \vec{a} = \vec{b} \times \vec{a} \) and \( \vec{r} \times \vec{b} = \vec{a} \times \vec{b} \), we will follow these steps:

### Step 1: Define the vectors
We have:
\[ \vec{a} = \hat{i} + \hat{j} \]
\[ \vec{b} = 2\hat{i} - \hat{k} \]

### Step 2: Set up the equations
From the first equation \( \vec{r} \times \vec{a} = \vec{b} \times \vec{a} \), we can express \( \vec{r} \) in terms of \( \vec{b} \) and \( \vec{a} \):
\[ \vec{r} = \vec{b} + \lambda \vec{a} \]
where \( \lambda \) is a scalar.

From the second equation \( \vec{r} \times \vec{b} = \vec{a} \times \vec{b} \), we can express \( \vec{r} \) in terms of \( \vec{a} \) and \( \vec{b} \):
\[ \vec{r} = \vec{a} + \mu \vec{b} \]
where \( \mu \) is another scalar.

### Step 3: Equate the two expressions for \( \vec{r} \)
Now we have two expressions for \( \vec{r} \):
\[ \vec{b} + \lambda \vec{a} = \vec{a} + \mu \vec{b} \]

### Step 4: Rearranging the equation
Rearranging gives us:
\[ \lambda \vec{a} - \mu \vec{b} = \vec{a} - \vec{b} \]

### Step 5: Compare coefficients
Now we can compare coefficients of \( \vec{a} \) and \( \vec{b} \):
- Coefficient of \( \vec{a} \): \( \lambda = 1 \)
- Coefficient of \( \vec{b} \): \( -\mu = -1 \) implies \( \mu = 1 \)

### Step 6: Substitute back to find \( \vec{r} \)
Substituting \( \lambda = 1 \) into \( \vec{r} = \vec{b} + \lambda \vec{a} \):
\[ \vec{r} = \vec{b} + \vec{a} \]
Now substituting the values of \( \vec{a} \) and \( \vec{b} \):
\[ \vec{r} = (2\hat{i} - \hat{k}) + (\hat{i} + \hat{j}) = (2\hat{i} + \hat{i}) + \hat{j} - \hat{k} = 3\hat{i} + \hat{j} - \hat{k} \]

### Step 7: Write the final point of intersection
Thus, the point of intersection is:
\[ (3, 1, -1) \]

### Final Answer
The point of intersection of the lines is \( (3, 1, -1) \), which corresponds to option (B).

---