Cross Product Math Example 1

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Example 1

medium
Find ⟨1,0,0⟩×⟨0,1,0⟩\langle 1, 0, 0 \rangle \times \langle 0, 1, 0 \rangle.

Solution

  1. 1
    Step 1: Use the formula: a×b=⟨a2b3−a3b2,a3b1−a1b3,a1b2−a2b1⟩\mathbf{a} \times \mathbf{b} = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle.
  2. 2
    Step 2: =⟨0⋅0−0⋅1,0⋅0−1⋅0,1⋅1−0⋅0⟩= \langle 0 \cdot 0 - 0 \cdot 1, 0 \cdot 0 - 1 \cdot 0, 1 \cdot 1 - 0 \cdot 0 \rangle.
  3. 3
    Step 3: =⟨0,0,1⟩= \langle 0, 0, 1 \rangle.
  4. 4
    Check: i^×j^=k^\hat{\mathbf{i}} \times \hat{\mathbf{j}} = \hat{\mathbf{k}} by the right-hand rule ✓

Answer

⟨0,0,1⟩\langle 0, 0, 1 \rangle
The cross product of two vectors gives a third vector perpendicular to both. For the standard basis vectors, i^×j^=k^\hat{\mathbf{i}} \times \hat{\mathbf{j}} = \hat{\mathbf{k}} follows the right-hand rule.

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}.

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More Cross Product Examples

Example 2 hard

Find [formula].

Example 3 medium

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Example 4 easy

Is [formula]?

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