Vector Dot Product : First Principles

Overview

**Vector Dot Product: First Principles**

» multiplied with component in parallel

→ $\overrightarrow{p}\cdot \overrightarrow{q}$ $=\overrightarrow{p}\cdot \overrightarrow{a}$ $=\left|\overrightarrow{p}\right|\left|\overrightarrow{a}\right|$

→ product of magnitudes of components in parallel

in parallel

We have developed the understanding that

• In some application scenarios, effect will be in the same direction as that of the cause.

• the observed effect can be sum of the effects of multiple different causes.

A vector can be split into components in parallel and in perpendicular directions to another vector.

Consider that force $\overrightarrow{f}}_{1$ is acting on an object which moves displacement $\overrightarrow{s}}_{r$. The effect $\overrightarrow{s}}_{r$ is split into components $\overrightarrow{s}}_{a$ and $\overrightarrow{s}}_{b$, where $\overrightarrow{s}}_{a$ is in the same direction as the cause $\overrightarrow{f}}_{1$. The work done by cause $f}_{1$ is '$f}_{1$ multiplied by $s}_{a$' as the effect is in the same direction and in that direction no other cause is observed.

Generalizing that... For two vectors $\overrightarrow{p}$ and $\overrightarrow{q}$, the **vector dot product** of the vectors is given by multiplication of

• magnitude of one vector and

• the magnitude of the component of second vector in parallel to the first vector.

$\overrightarrow{p}\cdot \overrightarrow{q}=\left|p\right|\left|a\right|$

where $\left|a\right|$ is the component of $\overrightarrow{q}$ in the direction of $\overrightarrow{p}$.

The component $a$ can be computed as '$\left|q\right|\mathrm{cos}\theta$', using basic trigonometry.

This product is formally defined as **vector dot product**.

$\overrightarrow{p}\cdot \overrightarrow{q}=\left|p\right|\left|q\right|\mathrm{cos}\theta$

Note that the result is a scalar. So, this operation is also called ** Scalar Product** of vectors.

A vector multiplication for components in parallel (in the same direction) is defined as scalar product or dot product.

The component in perpendicular does not take part in the dot product. This product of vector models application with the cause-effect relations that always in the same direction.

When two vector quantities interact to form a product, either one of the (1) component in parallel or (2) component in perpendicular is involved in the multiplication.

A vector multiplication for components in perpendicular is defined as cross product. This is explained later.

examples

Two vectors $\overrightarrow{p}$ and $\overrightarrow{q}$ with magnitudes $2$ and $3$ respectively are at an angle $60}^{\circ$. What is $\overrightarrow{p}\cdot \overrightarrow{q}$?

$\overrightarrow{p}\cdot \overrightarrow{q}$

$=\left|p\right|\left|q\right|\mathrm{cos}\theta$

$=2\times 3\times {\mathrm{cos}60}^{\circ}$

$=2\times 3\times \frac{1}{2}$

$=3$

Vector with magnitudes $4$ and $6$ are in the opposite directions. What is the dot product of these two vectors?

The answer is '$-24$'. The angle between the vectors is $180}^{\circ$ and ${\mathrm{cos}180}^{\circ}=-1$.

A vector $\overrightarrow{p}$ of magnitude $30$ is the cause. The observed effect is $\overrightarrow{q}$ of magnitude $20$ at an angle $45}^{\circ$ to the $\overrightarrow{p}$. What is the product between them?

The answer is '$300\sqrt{2}$'. Product between cause and effect is vector dot product $\left|p\right|\left|q\right|\mathrm{cos}\theta$.

summary

**Vector Dot Product / Scalar Product : ** is defined as

$\overrightarrow{p}\cdot \overrightarrow{q}=\left|p\right|\left|q\right|\mathrm{cos}\theta$ where $\theta$ is the angle between $\overrightarrow{p}$ and $\overrightarrow{q}$.

It is the product of components in parallel (i.e. in the same direction).

Outline

The outline of material to learn vector-algebra is as follows.

Note: Click here for detailed outline of vector-algebra.

• Introduction to Vectors

→ __Introducing Vectors__

→ __Representation of Vectors__

• Basic Properties of Vectors

→ __Magnitude of Vectors__

→ __Types of Vectors__

→ __Properties of Magnitude__

• Vectors & Coordinate Geometry

→ __Vectors & Coordinate Geometry__

→ __Position Vector of a point__

→ __Directional Cosine__

• Role of Direction in Vector Arithmetics

→ __Vector Arithmetics__

→ __Understanding Direction of Vectors__

• Vector Addition

→ __Vector Additin : First Principles__

→ __Vector Addition : Component Form__

→ __Triangular Law__

→ __Parallelogram Law__

• Multiplication of Vector by Scalar

→ __Scalar Multiplication__

→ __Standard Unit Vectors__

→ __Vector as Sum of Vectors__

→ __Vector Component Form__

• Vector Dot Product

→ __Introduction to Vector Multiplication __

→ __Cause-Effect-Relation__

→ __Dot Product : First Principles__

→ __ Dot Product : Projection Form__

→ __ Dot Product : Component Form__

→ __Dot Product With Direction__

• Vector Cross Product

→ __Vector Multiplication : Cross Product __

→ __Cross Product : First Principles__

→ __Cross Product : Area of Parallelogram__

→ __Cross Product : Component Form__

→ __Cross Product : Direction Removed__