Key Concepts:
Vector Representation:
A vector can be written in component form \( \mathbf{v} = (x, y) \), or as a directed line segment from one point to another.Vector Addition:
\[ \mathbf{a} + \mathbf{b} = \begin{pmatrix} a_x + b_x \\ a_y + b_y \end{pmatrix} \]Vector Subtraction:
\[ \mathbf{a} - \mathbf{b} = \begin{pmatrix} a_x - b_x \\ a_y - b_y \end{pmatrix} \]Multiplication by a Scalar:
\[ k\mathbf{v} = \begin{pmatrix} kx \\ ky \end{pmatrix} \]Magnitude of a Vector:
The length of a vector is given by: \[ |\mathbf{v}| = \sqrt{x^2 + y^2} \]Unit Vector:
A vector with magnitude 1, given by: \[ \hat{\mathbf{v}} = \frac{\mathbf{v}}{|\mathbf{v}|} \]Dot Product:
The dot product of two vectors is: \[ \mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y \]Direction of a Vector:
The angle \( \theta \) a vector makes with the positive x-axis is: \[ \theta = \tan^{-1} \left(\frac{y}{x}\right) \]Examples:
Example 1: Adding Two Vectors
Given \( \mathbf{a} = (3, 4) \) and \( \mathbf{b} = (-2, 1) \), find \( \mathbf{a} + \mathbf{b} \).Solution:
\[ \mathbf{a} + \mathbf{b} = \begin{pmatrix} 3 + (-2) \\ 4 + 1 \end{pmatrix} = \begin{pmatrix} 1 \\ 5 \end{pmatrix} \] Thus, the sum is \( \boxed{(1,5)} \).Example 2: Finding the Magnitude of a Vector
Find the magnitude of the vector \( \mathbf{v} = (5, -12) \).Solution:
\[ |\mathbf{v}| = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \] Thus, the magnitude is \( \boxed{13} \).Example 3: Finding the Unit Vector
Find the unit vector in the direction of \( \mathbf{v} = (6, 8) \).Solution:
Step 1: Find the magnitude: \[ |\mathbf{v}| = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \] Step 2: Compute the unit vector: \[ \hat{\mathbf{v}} = \frac{1}{10} \begin{pmatrix} 6 \\ 8 \end{pmatrix} = \begin{pmatrix} 0.6 \\ 0.8 \end{pmatrix} \] Thus, the unit vector is \( \boxed{(0.6, 0.8)} \).Example 4: Angle of a Vector
Find the angle \( \theta \) of the vector \( \mathbf{v} = (4, 3) \) with the x-axis.Solution:
\[ \theta = \tan^{-1} \left(\frac{3}{4} \right) \] Using a calculator: \[ \theta = \tan^{-1}(0.75) \approx 36.87^\circ \] Thus, the angle is \( \boxed{36.87^\circ} \).