Q.Find $|\vec{a} \times \vec{b}|$, if $\vec{a}=\hat{i}-7\hat{j}+7\hat{k}$ and $\vec{b}=3\hat{i}-2\hat{j}+2\hat{k}$.

Compute the cross product with the determinant, then take its length.

Determinant.

Expand.

Magnitude.

The idea

The magnitude of a cross product equals the area of the parallelogram the two vectors span. You could use $|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta$, but that needs the angle. When the components are given, it is far cleaner to build $\vec{a}\times\vec{b}$ from the determinant and then take its length.

Step-by-step

1. Write the vectors.

2. Set up the determinant.

3. Expand along the top row, remembering the middle term carries a minus sign:

• $\hat{i}$: $(-7)(2)-(7)(-2) = -14+14 = 0$
• $\hat{j}$: $-\big[(1)(2)-(7)(3)\big] = -\big[2-21\big] = 19$
• $\hat{k}$: $(1)(-2)-(-7)(3) = -2+21 = 19$

So

4. Take the magnitude.

Method: Magnitude of a Cross Product from Components

When both vectors are given in component form, build the cross product with the determinant and then take its length — no angle needed.

Steps

Step 1: Set up the determinant.

Step 2: Expand along the top row, minding the middle sign.

The $\hat{j}$ term carries a minus: $\hat{i}(a_2b_3-a_3b_2)-\hat{j}(a_1b_3-a_3b_1)+\hat{k}(a_1b_2-a_2b_1)$.

Step 3: Take the magnitude.

With the result $c_1\hat{i}+c_2\hat{j}+c_3\hat{k}$, compute $|\vec{a}\times\vec{b}|=\sqrt{c_1^2+c_2^2+c_3^2}$.

Common Mistakes

Mistake 1: Forgetting the minus sign on the $\hat{j}$ term.

Why it's wrong: the cofactor expansion makes the middle term $-\hat{j}(a_1b_3-a_3b_1)$; here the minor is $-19$, so the component is $+19$. Missing the sign flips it. Correct approach: keep the $-\hat{j}$ in the expansion.

Mistake 2: Confusing cross product with dot product.

Why it's wrong: $|\vec{a}\times\vec{b}|$ needs the vector (determinant) product, not $\vec{a}\cdot\vec{b}$. Correct approach: build $\vec{a}\times\vec{b}$ first, then take its length.

Mistake 3: Stopping at the vector $\vec{a}\times\vec{b}$.

Why it's wrong: the question asks for the magnitude. Correct approach: compute $\sqrt{0^2+19^2+19^2}=19\sqrt2$.