Skip to content
MathSolver Pro
  • Open the math solver
  • Ask the AI tutor
  • Find an AP course
  • Create a quiz
  • Open the calculator
  • Graphing Calculator
  • Browse all calculators
  • Read math guides
↑ ↓Enter to open · Esc to close
Solve a problem
VectorsLinear algebraCollegeabout 0:56

Find the cross product of ⟨0, 2, 5⟩ and ⟨1, -1, 3⟩: animated lesson

A narrated, step-by-step vectors video lesson with captions. Final answer: u × v = ⟨11, 5, -2⟩.

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.4 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    Find the cross product of ⟨0, 2, 5⟩ and ⟨1, -1, 3⟩.
  2. 2. Steps, part 1
    1. u×v=∣ijk0251−13∣u \times v = \begin{vmatrix} \mathbf i & \mathbf j & \mathbf k \\ 0 & 2 & 5 \\ 1 & -1 & 3 \end{vmatrix}
      Write the determinant with unit vectors i, j, k in the first row
    2. (2)(3)−(5)(−1)=11(2)(3) - (5)(-1) = 11
      i component
    3. −((0)(3)−(5)(1))=5-\big((0)(3) - (5)(1)\big) = 5
      j component (note the minus sign in the expansion)
    4. (0)(−1)−(2)(1)=−2(0)(-1) - (2)(1) = -2
      k component
  3. 3. Steps, part 2
    1. ∣u×v∣=150=56≈12.24744871|u \times v| = \sqrt{150} = 5\sqrt{6} \approx 12.24744871
      Magnitude = area of the parallelogram spanned by u and v
    2. (u×v)u=0,(u×v)v=0(u\times v) u = 0,\quad (u\times v) v = 0
      Check orthogonality
  4. 4. The answer
    u×v=⟨11,5,−2⟩u \times v = \langle 11, 5, -2 \rangle

    u×v = ⟨11, 5, -2⟩

  5. 5. Check
    • The result is perpendicular to both input vectors (both dot products are 0).

Transcript

The problem. Here is the problem: Find the cross product of ⟨0, 2, 5⟩ and ⟨1, negative 1, 3⟩. Cross product of two 3D vectors.

Steps, part 1. Write the determinant with unit vectors i, j, k in the first row. i component. j component (note the minus sign in the expansion). k component.

Steps, part 2. Magnitude equals area of the parallelogram spanned by u and v. Check orthogonality.

The answer. So the answer is u times v equals ⟨11, 5, negative 2⟩. The result is perpendicular to both u and v; its length 5the square root of 6 approximately 12.24744871 is the area of the parallelogram they span (the triangle has half that area).

Check. The result is perpendicular to both input vectors (both dot products are 0).

About vectors

Magnitude, unit vectors, dot products and angles between vectors.

More lessons like this

Find the cross product of ⟨0, 2, 5⟩ and ⟨1, -1, 3⟩ (Lesson)