🧭 Linear Algebra & Kinematics
2D & 3D Vector Calculator
Compute vector Dot Product ($\vec{u} \cdot \vec{v}$), Cross Product ($\vec{u} \times \vec{v}$), Euclidean lengths, unit vectors ($\hat{u}$), and angular separation ($\theta$).
Vector Components
Dot Product ($\vec{u} \cdot \vec{v}$)
56.00
Cross Product ($\vec{u} \times \vec{v}$)
⟨-3, 6, -3⟩
Vector Sum ($\vec{u} + \vec{v}$)
⟨7, 9, 11⟩
Magnitude $|\vec{u}|$
5.3852
Magnitude $|\vec{v}|$
10.4881
💡 Unit Vector Normalization ($\hat{u}$):
⟨0.3714, 0.5571, 0.7428⟩
Vector Algebra Fundamentals
Vectors represent quantities with both magnitude and directional orientation in Euclidean space:
- Dot Product (Scalar): $$\vec{u} \cdot \vec{v} = u_x v_x + u_y v_y + u_z v_z = |\vec{u}| |\vec{v}| \cos\theta$$
- Cross Product (Vector): $$\vec{u} \times \vec{v} = \langle u_y v_z - u_z v_y, \; u_z v_x - u_x v_z, \; u_x v_y - u_y v_x \rangle$$