Q.Find the angle between the lines $\vec{r} = 3\hat{i} - 2\hat{j} + 6\hat{k} + \lambda(2\hat{i} + \hat{j} + 2\hat{k})$ and $\vec{r} = (2\hat{j} - 5\hat{k}) + \mu(6\hat{i} + 3\hat{j} + 2\hat{k})$.

Concept: Angle Between Lines — the angle between two lines equals the angle between their direction vectors.

Step 1: Identify direction vectors.

For the first line: $\vec{b}_1 = 2\hat{i} + \hat{j} + 2\hat{k}$.

For the second line: $\vec{b}_2 = 6\hat{i} + 3\hat{j} + 2\hat{k}$.

Step 2: Use the dot product formula:

$\cos\theta = \frac{\vec{b}_1 \cdot \vec{b}_2}{|\vec{b}_1|\,|\vec{b}_2|}$.

Compute:

$\vec{b}_1 \cdot \vec{b}_2 = (2)(6) + (1)(3) + (2)(2) = 12 + 3 + 4 = 19$. …

The key idea: two lines in vector form $\vec{r} = \vec{a} + \lambda \vec{b}$ are defined by a fixed point $\vec{a}$ and a direction vector $\vec{b}$. The angle between the lines is simply the angle between their direction vectors — the position vectors $\vec{a}$ don't matter because lines can be shifted without changing their orientation.

So we ignore $\vec{a}_1 = 3\hat{i} - 2\hat{j} + 6\hat{k}$ and $\vec{a}_2 = 2\hat{j} - 5\hat{k}$ entirely. Only the direction vectors matter.

1. Identify the direction vectors.

   From the first line: $\vec{b}_1 = 2\hat{i} + \hat{j} + 2\hat{k}$.

   From the second line: $\vec{b}_2 = 6\hat{i} + 3\hat{j} + 2\hat{k}$.

2. Recall the formula for the angle between two vectors.

   If $\theta$ is the angle between $\vec{b}_1$ and $\vec{b}_2$, then

This comes directly from the dot product definition: $\vec{b}_1 \cdot \vec{b}_2 = |\vec{b}_1||\vec{b}_2|\cos\theta$.

3. Compute the dot product.

4. Compute the magnitudes.

5. Put it together. …

Method: Angle between two lines from their direction vectors

Use this for the angle between two lines given in vector or Cartesian form.

Steps

Step 1: Extract only the direction vectors.

From $\vec{r}=\vec{a}+\lambda\vec{b}$, the direction is $\vec{b}$ (the $\lambda$-coefficients). The position vectors $\vec{a}$ play NO role in the angle — a line can be slid without changing its orientation.

Step 2: Apply the cosine formula with a modulus.

$\cos\theta=\frac{|\vec{b}_1\cdot\vec{b}_2|}{|\vec{b}_1|\,|\vec{b}_2|}.$ …

Common Mistakes

Mistake 1: Including the position vectors in the dot product.

Why it's wrong: $\vec{a}_1$ and $\vec{a}_2$ only locate the lines; the angle depends solely on the directions $\vec{b}_1,\vec{b}_2$. Correct approach: use only $(2,1,2)$ and $(6,3,2)$.

Mistake 2: Dropping the modulus in the numerator. …