Vector Product (Cross Product) of Two Vectors

Why we care

The “cross” of two vectors helps us build moment of force (torque) and angular momentum—two pillars of rotational motion.:contentReference[oaicite:0]{index=0}

Definition

  • For two vectors \(\mathbf a\) and \(\mathbf b\), their vector product is \(\mathbf c = \mathbf a \times \mathbf b\).
  • Magnitude: \( |\mathbf c| = ab\sin\theta \), where \(a\) and \(b\) are the lengths of the two vectors and \(\theta\) is the smaller angle between them.:contentReference[oaicite:1]{index=1}
  • Direction: \(\mathbf c\) stands straight out of the plane of \(\mathbf a\) and \(\mathbf b\). Point your right-hand fingers from \(\mathbf a\) toward \(\mathbf b\); your thumb shows \(\mathbf c\).:contentReference[oaicite:2]{index=2}

Key properties

  • Perpendicularity: \(\mathbf c\) is perpendicular to both \(\mathbf a\) and \(\mathbf b\).:contentReference[oaicite:3]{index=3}
  • Non-commutative: \(\mathbf a \times \mathbf b = -(\mathbf b \times \mathbf a)\). Same size, opposite direction.:contentReference[oaicite:4]{index=4}
  • Distributive: \(\mathbf a \times (\mathbf b+\mathbf c)=\mathbf a \times \mathbf b+\mathbf a \times \mathbf c\).:contentReference[oaicite:5]{index=5}
  • Self-product vanishes: \(\mathbf a \times \mathbf a=\mathbf 0\).:contentReference[oaicite:6]{index=6}
  • Mirror-safe: reflecting the whole system flips each vector but not their cross product.:contentReference[oaicite:7]{index=7}

Right-hand “cheat sheet” for unit vectors

Follow the cyclic order \(\hat{\mathbf i}\rightarrow\hat{\mathbf j}\rightarrow\hat{\mathbf k}\rightarrow\hat{\mathbf i}\):

  • \(\hat{\mathbf i}\times\hat{\mathbf j}= \hat{\mathbf k}\)
  • \(\hat{\mathbf j}\times\hat{\mathbf k}= \hat{\mathbf i}\)
  • \(\hat{\mathbf k}\times\hat{\mathbf i}= \hat{\mathbf j}\)
  • Reverse the order and you pick up a minus sign, e.g., \(\hat{\mathbf j}\times\hat{\mathbf i}= -\hat{\mathbf k}\).:contentReference[oaicite:8]{index=8}

Component formula (determinant form)

\[ \mathbf a \times \mathbf b= \begin{vmatrix} \hat{\mathbf i}&\hat{\mathbf j}&\hat{\mathbf k}\\ a_x & a_y & a_z\\ b_x & b_y & b_z \end{vmatrix} \] This is the quickest way to crank out the cross product once you know the components.:contentReference[oaicite:9]{index=9}

Worked idea (outline)

With \(\mathbf a=3\hat{\mathbf i}+4\hat{\mathbf j}+5\hat{\mathbf k}\) and \(\mathbf b=-2\hat{\mathbf i}+\,1\hat{\mathbf j}+3\hat{\mathbf k}\):

  • Dot product: \(\mathbf a\cdot\mathbf b=-25\).
  • Cross product: \(\mathbf a \times \mathbf b = 7\hat{\mathbf i}+5\hat{\mathbf j}-11\hat{\mathbf k}\).:contentReference[oaicite:10]{index=10}

(The numbers show how the determinant method works in practice.)

NEET high-yield ideas

  1. Magnitude formula \(ab\sin\theta\) and the insistence on the smaller angle.
  2. Right-hand rule—curl from \(\mathbf a\) to \(\mathbf b\); thumb is \(\mathbf a\times\mathbf b\).
  3. Non-commutative nature: sign flip when you swap vectors.
  4. Unit-vector cross products (\(\hat{\mathbf i},\hat{\mathbf j},\hat{\mathbf k}\)) and the cyclic rule.
  5. Determinant shortcut for quick component calculation.

Keep practicing the right-hand trick—it turns vector puzzles into simple thumb rules!