Vector Calculator

Understanding Vector Operations

Vectors are fundamental mathematical objects that have both magnitude and direction. They are used extensively in physics, engineering, computer graphics, and many other fields. This calculator helps you perform common vector operations:

• Dot Product (Scalar Product): \(\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta)\). Results in a scalar. Useful for finding the angle between vectors and projections.
• Cross Product (Vector Product): \(ec{a} imes ec{b}\). Results in a vector perpendicular to both \(\vec{a}\) and \(\vec{b}\). Defined in 3D only, useful for finding area and torque.
• Vector Addition/Subtraction: Component-wise operations. \(ec{a} + ec{b} = (a_x+b_x, a_y+b_y, a_z+b_z)\).
• Angle Between Vectors: Calculated using the dot product formula: \(\theta = \cos^{-1} \left( \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|} \right)\).
• Projection of \(\vec{a}\) onto \(ec{b}\): The vector component of \(\vec{a}\) in the direction of \(\vec{b}\). Formula: \(\text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2} \vec{b}\).
• Magnitude (Norm) of a Vector: Length of the vector. \(|\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}\).
• Normalization of a Vector: Converting a vector into a unit vector (magnitude 1) in the same direction. \(\hat{a} = \frac{\vec{a}}{|\vec{a}|}\).

Use this tool to quickly perform these calculations and deepen your understanding of vector mathematics. For more in-depth information, refer to linear algebra textbooks or online resources like Wikipedia's Vector article.