Cross Product Math Example 2

Follow the full solution, then compare it with the other examples linked below.

Example 2

hard
Find ⟨2,3,1⟩×⟨1,−1,2⟩\langle 2, 3, 1 \rangle \times \langle 1, -1, 2 \rangle.

Solution

  1. 1
    Step 1: ii-component: 3(2)−1(−1)=6+1=73(2) - 1(-1) = 6 + 1 = 7.
  2. 2
    Step 2: jj-component: 1(1)−2(2)=1−4=−31(1) - 2(2) = 1 - 4 = -3.
  3. 3
    Step 3: kk-component: 2(−1)−3(1)=−2−3=−52(-1) - 3(1) = -2 - 3 = -5.
  4. 4
    Result: ⟨7,−3,−5⟩\langle 7, -3, -5 \rangle.
  5. 5
    Check: ⟨2,3,1⟩⋅⟨7,−3,−5⟩=14−9−5=0\langle 2,3,1 \rangle \cdot \langle 7,-3,-5 \rangle = 14 - 9 - 5 = 0 ✓ (perpendicular)

Answer

⟨7,−3,−5⟩\langle 7, -3, -5 \rangle
The cross product is computed component by component using the cyclic pattern. The check that both original vectors are perpendicular to the result (dot product = 0) verifies correctness.

About Cross Product

The cross product of two 3D vectors a=⟨a1,a2,a3⟩\mathbf{a} = \langle a_1, a_2, a_3 \rangle and b=⟨b1,b2,b3⟩\mathbf{b} = \langle b_1, b_2, b_3 \rangle is a new vector a×b\mathbf{a} \times \mathbf{b} that is perpendicular to both a\mathbf{a} and b\mathbf{b}. Its magnitude equals the area of the parallelogram formed by a\mathbf{a} and b\mathbf{b}.

Learn more about Cross Product →

More Cross Product Examples

Example 1 medium

Find [formula].

Example 3 medium

Find [formula].

Example 4 easy

Is [formula]?

← All Cross Product Examples Cross Product Concept