# [[Dot Product of Perpendicular Vectors]] (Mathematics > Vector Algebra) ## Definition Two nonzero vectors $\mathbf{a}$ and $\mathbf{b}$ are **perpendicular** (or orthogonal) if the angle $\theta$ between them is $90^\circ$. This relationship is often denoted by the symbol $\perp$ as: $ \mathbf{a} \perp \mathbf{b} $ In this case, their dot product is zero. ## Key Concept - **Orthogonality and Dot Product**: For perpendicular vectors $\mathbf{a}$ and $\mathbf{b}$ with $\theta = 90^\circ$, we have: $ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| \cdot |\mathbf{b}| \cdot \cos 90^\circ = |\mathbf{a}| \cdot |\mathbf{b}| \cdot 0 = 0 $ ## Important Properties 1. **Zero Dot Product Condition**: Two nonzero vectors are perpendicular if and only if their dot product is zero: $ \mathbf{a} \perp \mathbf{b} \iff \mathbf{a} \cdot \mathbf{b} = 0 $ 2. **Converse is True**: If $\mathbf{a} \cdot \mathbf{b} = 0$ and both vectors are nonzero, then $\mathbf{a}$ and $\mathbf{b}$ are perpendicular. ## Core Example 1. **Determine Orthogonality**: Given $\mathbf{a} = \langle 3, 4 \rangle$ and $\mathbf{b} = \langle -4, 3 \rangle$, check if $\mathbf{a}$ and $\mathbf{b}$ are perpendicular. - Calculate $\mathbf{a} \cdot \mathbf{b}$: $ \mathbf{a} \cdot \mathbf{b} = (3)(-4) + (4)(3) = -12 + 12 = 0 $ - Since $\mathbf{a} \cdot \mathbf{b} = 0$, $\mathbf{a}$ and $\mathbf{b}$ are perpendicular. ## Related Topics - [[Odd Functions]] - [[Rotation in a Plane]] - [[Rotation by 270 Degrees]] - [[Rotational Symmetry of Regular Polygons]] - [[Area of a Triangle Using Trigonometry]] - [[Reflection Across the Line y = -x]] - [[Effect of Multiplying by \(i\)]] - [[Rotation by Multiplying with \(i\)]] - [[Perpendicular Distance from a Point to a Line]] - [[Distance from a Point to a Line]] - [[Perpendicular Slopes]] - [[30-60-90 Triangle]] - [[Dot Product]] - [[Vector Magnitude]] - [[Orthogonal Vectors]] ## Quick Review Questions 1. Why does the dot product of two perpendicular vectors equal zero? 2. Given vectors $\mathbf{a} = \langle 1, 2, 3 \rangle$ and $\mathbf{b} = \langle 3, -6, 2 \rangle$, determine if they are perpendicular. *** ![[Dot_Product_of_Perpendicular_Vectors_visualization]]