This may seem slightly counter-intuitive at first. But if we add in some possible figures you can see how it works. If moves 5 units to the left and moves 1 unit to the right (−left) and 3 units down.
Then units to the left −1 unit to the right and 3 units down = 4 units to the left and 3 units down.
Frequently Asked Questions About Vectors in Mathematics
In mathematics and physics, a vector is a quantity that has both magnitude (or length) and direction.
Examples of vector quantities include displacement, velocity, acceleration, and force. Unlike scalar quantities (like mass, temperature, or time) which only have magnitude, vectors tell us "how much" and "in which direction".
Graphically, a vector is often represented by an arrow. The length of the arrow represents the magnitude, and the arrow points in the direction of the vector.
Vectors can be represented in several ways:
- Component Form: In a coordinate system (like 2D or 3D Cartesian), a vector can be represented by its components along the axes. For example, in 2D, a vector starting from the origin might be written as \(\langle x, y \rangle\) or \((x, y)\), where \(x\) and \(y\) are its components in the x and y directions. In 3D, it would be \(\langle x, y, z \rangle\).
- Column/Row Matrix: \(\begin{pmatrix} x \\ y \end{pmatrix}\) or \(\begin{pmatrix} x & y \end{pmatrix}\).
- Using Basis Vectors: Vectors can be written as a sum of scalar multiples of basis vectors (unit vectors along the axes). In 2D, using standard basis vectors \(\mathbf{i} = \langle 1, 0 \rangle\) and \(\mathbf{j} = \langle 0, 1 \rangle\), the vector \(\langle x, y \rangle\) is \(x\mathbf{i} + y\mathbf{j}\). In 3D, using \(\mathbf{i}, \mathbf{j}, \mathbf{k}\), it's \(x\mathbf{i} + y\mathbf{j} + z\mathbf{k}\).
The easiest way to add or subtract vectors mathematically is using their component forms:
- Addition: To add two vectors, add their corresponding components.
If \(\mathbf{u} = \langle x_1, y_1 \rangle\) and \(\mathbf{v} = \langle x_2, y_2 \rangle\), then
\(\mathbf{u} + \mathbf{v} = \langle x_1 + x_2, y_1 + y_2 \rangle\).
(This extends to 3D or more dimensions). - Subtraction: To subtract one vector from another, subtract their corresponding components.
If \(\mathbf{u} = \langle x_1, y_1 \rangle\) and \(\mathbf{v} = \langle x_2, y_2 \rangle\), then
\(\mathbf{u} - \mathbf{v} = \langle x_1 - x_2, y_1 - y_2 \rangle\).
Graphically, vector addition can be visualized using the "tip-to-tail" method or the "parallelogram" method.
The magnitude of a vector is its length. If the vector is in component form, you can find its magnitude using a generalization of the Pythagorean theorem.
- For a 2D vector \(\mathbf{v} = \langle x, y \rangle\), the magnitude (denoted by \(\|\mathbf{v}\|\) or \(|\mathbf{v}|\)) is:
\(\|\mathbf{v}\| = \sqrt{x^2 + y^2}\) - For a 3D vector \(\mathbf{v} = \langle x, y, z \rangle\), the magnitude is:
\(\|\mathbf{v}\| = \sqrt{x^2 + y^2 + z^2}\)
The magnitude is always a non-negative scalar value.
Vector Addition Calculator (2D)
Enter the components of two 2D vectors \(\mathbf{u} = \langle x_1, y_1 \rangle\) and \(\mathbf{v} = \langle x_2, y_2 \rangle\) to find their sum \(\mathbf{u} + \mathbf{v}\).
