Vector Operations Calculator

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Example

Created on 2024-06-20Asked by Samuel Wright (Solvelet student)

Compute the dot product and cross product of the vectors

\mathbf{a} = \langle 1, 2, 3 \rangle

and

\mathbf{b} = \langle 4, 5, 6 \rangle

Solution

To compute the dot product and cross product of the vectors

\mathbf{a} = \langle 1, 2, 3 \rangle

and

\mathbf{b} = \langle 4, 5, 6 \rangle

: 1. **Dot product:**

\mathbf{a} \cdot \mathbf{b} = 1 \cdot 4 + 2 \cdot 5 + 3 \cdot 6 = 4 + 10 + 18 = 32.

2. **Cross product:**

\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 2 & 3 \\ 4 & 5 & 6 \end{vmatrix}.

3. **Calculate the determinant:**

\mathbf{a} \times \mathbf{b} = \mathbf{i}(2 \cdot 6 - 3 \cdot 5) - \mathbf{j}(1 \cdot 6 - 3 \cdot 4) + \mathbf{k}(1 \cdot 5 - 2 \cdot 4) = \mathbf{i}(12 - 15) - \mathbf{j}(6 - 12) + \mathbf{k}(5 - 8).

\mathbf{a} \times \mathbf{b} = -3\mathbf{i} + 6\mathbf{j} - 3\mathbf{k}.

4. **Result:**

\mathbf{a} \cdot \mathbf{b} = 32, \quad \mathbf{a} \times \mathbf{b} = \langle -3, 6, -3 \rangle.

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Abigail Hall on Solvelet

1. Compute the dot product of the vectors

\langle 2, -1, 3 \rangle

and

\langle -3, 4, 1 \rangle

.2. Find the magnitude of the vector

\langle -1, 2, -2 \rangle

DefinitionThese operations include addition, subtraction, scalar multiplication, dot product, and cross product, and are used to perform tasks such as adding vectors together, subtracting vectors, scalar vector multiplication, dot product for dot product computation etc. in an effort to do vector manipulation in vector spaces. Example; a=⟨1,2⟩, b=⟨3,4⟩; (a.b)=1×3+2×4=11.

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