# [[Dot Product]] (Mathematics > Vector Algebra) ## Definition The dot product (or scalar product) of two vectors $\mathbf{a}$ and $\mathbf{b}$ is a scalar quantity defined by: $ \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}| \cdot |\mathbf{b}| \cdot \cos \theta, $ where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$ when their tails are positioned at the same point. ## Key Concepts - **Scalar Output**: The result of the dot product is a scalar, not a vector. - **Angle Dependency**: The dot product depends on the cosine of the angle between the two vectors. - **Orthogonal Vectors**: If $\theta = 90^\circ$, then $\cos \theta = 0$, so $\mathbf{a} \cdot \mathbf{b} = 0$ for perpendicular vectors. ## Important Properties 1. **Commutative Property**: $\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}$. 2. **Distributive Property**: $\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}$. 3. **Dot Product of a Vector with Itself**: $\mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2$. ## Essential Formulas - **Dot Product in Terms of Components**: If $\mathbf{a} = \langle a_1, a_2, a_3 \rangle$ and $\mathbf{b} = \langle b_1, b_2, b_3 \rangle$, then $ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3. $ ## Core Examples 1. **Example with Angles**: Given $\mathbf{a}$ and $\mathbf{b}$ with $|\mathbf{a}| = 3$, $|\mathbf{b}| = 4$, and $\theta = 60^\circ$, calculate $\mathbf{a} \cdot \mathbf{b}$. $ \mathbf{a} \cdot \mathbf{b} = 3 \cdot 4 \cdot \cos 60^\circ = 12 \cdot \frac{1}{2} = 6 $ 2. **Component Form Example**: Let $\mathbf{a} = \langle 1, 2, 3 \rangle$ and $\mathbf{b} = \langle 4, -1, 2 \rangle$. Then, $ \mathbf{a} \cdot \mathbf{b} = (1)(4) + (2)(-1) + (3)(2) = 4 - 2 + 6 = 8 $ ## Related Theorems/Rules - **Cauchy-Schwarz Inequality**: $|\mathbf{a} \cdot \mathbf{b}| \leq |\mathbf{a}| \cdot |\mathbf{b}|$, with equality if and only if $\mathbf{a}$ and $\mathbf{b}$ are linearly dependent. ## Common Pitfalls - **Confusing Dot and Cross Products**: Remember, the dot product yields a scalar, while the cross product results in a vector. - **Misinterpreting the Angle**: Ensure $\theta$ is the angle between the vectors, not an arbitrary angle. ## Related Topics - [[One-to-One Functions]] - [[Translation]] - [[Rotation in a Plane]] - [[Rotation by 270 Degrees]] - [[General Transformations of Functions]] - [[Perpendicular Distance from a Point to a Line]] - [[Distance from a Point to a Line]] - [[Perpendicular Slopes]] - [[30-60-90 Triangle]] - [[Cross Product]] - [[Vector Magnitude]] - [[Cauchy-Schwarz Inequality]] ## Quick Review Questions 1. What is the dot product of two perpendicular vectors, and why? 2. Calculate the dot product of $\mathbf{a} = \langle 3, 0, -2 \rangle$ and $\mathbf{b} = \langle 1, 4, 2 \rangle$. *** ![[Dot_Product_visualization]]