Q.The number of vectors of unit length perpendicular to the vectors $\vec{a}=2\hat{i}+2\hat{j}+\hat{k}$ and $\vec{b}=\hat{j}+\hat{k}$ is
(A) one
(B) two
(C) three
(D) infinite

The key idea is that any vector perpendicular to both $\vec{a}$ and $\vec{b}$ must be parallel to their cross product $\vec{a} \times \vec{b}$.

Step 1: Compute the cross product.

Step 2: The magnitude of this vector is

$|\vec{a} \times \vec{b}| = \sqrt{1^2 + (-2)^2 + 2^2} = \sqrt{9} = 3.$ …

We need vectors that are perpendicular to both given vectors at the same time. That is the geometric meaning of being perpendicular to two non-parallel vectors: the vector must lie along the line that is orthogonal to the plane containing $\vec{a}$ and $\vec{b}$. That line has exactly two unit-length directions — one pointing one way, the other pointing exactly opposite.

The condition for a vector $\vec{v}$ to be perpendicular to $\vec{a}$ is $\vec{v} \cdot \vec{a} = 0$, and similarly $\vec{v} \cdot \vec{b} = 0$. A vector satisfying both dot products is orthogonal to the plane spanned by $\vec{a}$ and $\vec{b}$. The cross product $\vec{a} \times \vec{b}$ is the natural vector that is perpendicular to both. So any vector perpendicular to both must be a scalar multiple of $\vec{a} \times \vec{b}$.

Now we just need the unit vectors along that direction.

1. Compute $\vec{a} \times \vec{b}$

   $\vec{a} = 2\hat{i} + 2\hat{j} + \hat{k}$, $\vec{b} = 0\hat{i} + \hat{j} + \hat{k}$.

Expand:

So $\vec{a} \times \vec{b} = \hat{i} - 2\hat{j} + 2\hat{k}$.

2. Find its magnitude

3. Unit vectors perpendicular to both

   The unit vector in the direction of $\vec{a} \times \vec{b}$ is:

   $\hat{n}_1 = \frac{\vec{a} \times \vec{b}}{|\vec{a} \times \vec{b}|} = \frac{1}{3}(\hat{i} - 2\hat{j} + 2\hat{k})$ …

Method: Counting unit vectors perpendicular to two given vectors

Use this whenever you must count (or find) unit vectors perpendicular to two given vectors at once.

Steps

Step 1: Recognise the perpendicular direction is the cross product

Any vector perpendicular to both $\vec{a}$ and $\vec{b}$ must lie along $\vec{a}\times\vec{b}$, which is by construction perpendicular to each.

Step 2: Check the two vectors are not parallel

If $\vec{a}\times\vec{b}=\vec{0}$ (parallel vectors), the perpendicular direction is not unique and there are infinitely many such unit vectors. Otherwise the direction is a single line. …

Common Mistakes

Mistake 1: Answering "one"

Why it's wrong: the direction $\vec a\times\vec b$ and its exact opposite $-(\vec a\times\vec b)$ are both unit-length and both perpendicular to $\vec a$ and $\vec b$. Correct approach: include the $\pm$, giving exactly two.

Mistake 2: Answering "infinite" for non-parallel vectors …