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Vector addition involves combining two or more vectors to produce a resultant vector. This is done by adding the corresponding components of each vector. For instance, let's consider vectors \(\textbf{A} = \langle 3,1 \rangle\) and \(\textbf{B} = \langle -2, 3 \rangle\). To find the sum, \(\textbf{A} + \textbf{B}\), simply add the x-components and the y-components separately.
So, \(\textbf{A} + \textbf{B} = \langle 3 + (-2), 1 + 3 \rangle = \langle 1, 4 \rangle\). This new vector represents moving 1 unit in the x-direction and 4 units in the y-direction.
By visualizing or plotting these vectors on a coordinate plane, you can better understand how they combine to form the resultant vector.

The magnitude of a vector represents its length, which can be calculated using the Pythagorean theorem. If you have a vector \(\textbf{v} = \langle x, y \rangle\), the magnitude is found using the formula: \[ \| \textbf{v} \| = \sqrt{x^2 + y^2} \] For our resultant vector \(\textbf{A} + \textbf{B} = \langle 1, 4 \rangle\), this formula becomes:
\[ \| \textbf{A} + \textbf{B} \| = \sqrt{1^2 + 4^2} = \sqrt{1 + 16} = \sqrt{17} \] The magnitude is approximately 4.12 (when rounded to two decimal places). This value shows the actual distance or length from the origin to the point (1, 4) on a coordinate plane.

The direction angle of a vector specifies the angle it makes with the positive x-axis. To find this angle for a given vector \(\textbf{v} = \langle x, y \rangle\), you use the inverse tangent function: \[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \] For our resultant vector \(\textbf{A} + \textbf{B} = \langle 1, 4 \rangle\), the calculation becomes:
\[ \theta = \tan^{-1} \left( \frac{4}{1} \right) = \tan^{-1}(4) \] This will give us the direction angle in either radians or degrees, depending on the context. For example, using a calculator, \(\tan^{-1}(4)\) approximately equals 75.96 degrees.
This angle reflects how steep the line of the vector \(\langle 1, 4 \rangle\) is relative to the positive x-axis.