Server Error

Server Not Reachable.

This may be due to your internet connection or the nubtrek server is offline.

Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.

User Guide

Welcome to nubtrek.

The content is presented in small-focused learning units to enable you to
think,
figure-out, &
learn.

Just keep tapping (or clicking) on the content to continue in the trail and learn.

User Guide

To make best use of nubtrek, understand what is available.

nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

nub,

trek,

jogger,

exercise.

User Guide

nub is the simple explanation of the concept.

This captures the small-core of concept in simple-plain English. The objective is to make the learner to think about.

User Guide

trek is the step by step exploration of the concept.

Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step.

User Guide

jogger provides the complete mathematical definition of the concepts.

This captures the essence of learning and helps one to review at a later point. The reference is available in pdf document too. This is designed to be viewed in a smart-phone screen.

User Guide

exercise provides practice problems to become fluent in the concepts.

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge.

summary of this topic

### Vector Dot Product

Voice

Voice

Home

Vector Dot Product : Component Form

»  sum of product of individual components
→  vec p cdot vec q = p_xq_x+p_yq_y+p_zq_z
→  vec p = p_x i+p_yj+p_zk
→  vec q = q_x i+q_yj+q_zk

### Vector Dot Product : Component Form

plain and simple summary

nub

plain and simple summary

nub

dummy

For given two vectors in component forms, the dot product is the sum of product of corresponding components of the vectors.

simple steps to build the foundation

trek

simple steps to build the foundation

trek

Support Nubtrek

You are learning the free content, however do shake hands with a coffee to show appreciation.
To stop this message from appearing, please choose an option and make a payment.

Keep tapping on the content to continue learning.
Starting on learning "component form of vector dot product". ;; In this page, you will learn about the component form of the vector dot product.

Given the two vectors
vec p = p_x i+p_yj+p_zk

vec q = q_x i+q_yj+q_zk
what will be the vector dot product?

• |p||q|cos theta
• |p||q|

answer is |p||q|cos theta.

How will one compute the angle theta from the given component forms of vectors
vec p = p_x i+p_yj+p_zk
vec q = q_x i+q_yj+q_zk

Consider this as triangle in coordinate plane.
vec p = p_x i+p_yj+p_zk
vec q = q_x i+q_yj+q_zk. The triangle is made of 3 sides having scalar quantities p=|vec p|, q=|vec q| and r = |vec q-vec p|. The cosine rule of triangle is applicable.
r^2=p^2+q^2-2pq cos theta

With algebraic manipulations on this, we can derive that
cos theta = (p_xq_x+p_yq_y+p_zq_z)/(|p||q|)
Substituting the above in the vector dot product we get.
vec p cdot vec q
quad quad = |p||q|cos theta
quad quad = |p||q|(p_xq_x+p_yq_y+p_zq_z)/(|p||q|)
quad quad = p_xq_x+p_yq_y+p_zq_z

That derives the component form of vector dot product as
vec p cdot vec q
quad quad = p_xq_x+p_yq_y+p_zq_z

This proof requires one to recall the cosine rule of triangles. A simpler proof, that a student can easily derive, is given in the coming pages.

Bilinear Property : For any vector vec p, vec q, vec r in bbb V and lambda in RR
(lambda vec p + vec q) cdot vec r = lambda (vec p cdot vec r) + (vec q cdot vec r)
This is explained and proven in properties of the dot product. For now, consider this to be true.

A vector vec p = p_x i + p_y j + p_z k is sum of scalar multiple of vectors. i, j, k are unit vectors, and the scalar multiples are p_x, p_y, p_z.

The same applies to vec q = q_x i + q_y j + q_z k Sum of multiple of vectors.

Proof for component form of vector dot product using bilinear property of dot product.

vec p cdot vec q
quad quad = (p_x i + p_y j + p_z k) cdot
quad quad quad quad (q_x i + q_y j + q_z k)
Apply bilinear property of dot product
quad quad = p_x i cdot (q_x i + q_y j + q_z k)
quad quad quad quad + p_y j cdot (q_x i + q_y j + q_z k)
quad quad quad quad + p_z k cdot (q_x i + q_y j + q_z k)
Apply i cdot i = i,j cdot j = j, k cdot k = k
i cdot j = 0, j cdot k = 0, k cdot i = 0
quad quad = p_xq_x+p_yq_y+p_zq_z

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Vector Dot Product in Component Form: For given two vectors vec p = p_x i+p_yj+p_zk and
vec q = q_x i+q_yj+q_zk

vec p cdot vec q = p_xq_x+p_yq_y+p_zq_z

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Given vec p = 2i+1.2j-k and vec q = i-j+k what is vec p cdot vec q?

• 2.2
• .2
• -.2
• 4.2

The answer is '-.2'

Progress

Progress

Given the two vectors p = p x i + p y j + p z k ;; vector q = q x i + q y j + q z k ;; what will be the vector dot product?
1
2
The answer is "magnitude of p multiplied magnitude of q multiplied cos theta". How will one compute the angle theta from the given component forms of the vectors p and q?
Consider this as triangle in coordinate plane. p = p x i + p y j + p z k ;; vector q = q x i + q y j + q z k ;; The triangle is made of 3 sides having scalar quantities p=magnitude of vector p, q = magnitude of vector q and r = vector q minus vector p. The cosine rule of triangle is applicable. r squared = p squared + q squared minus 2 p q cos theta. ;; With algebraic manipulations on this, we can derive that ;; cos theta = p x q x, + p y q y, + p z q z, divided by, magnitude of p magnitude of q. ;; Substituting the above in the vector dot product we get.;; vector p dot vector q = magnitude of p magnitude of q cos theta ;; equals p x q x, + p y q y, + p z q z.
That derives the component form of vector dot product as vector p dot vector q equals p x q x, + p y q y, + p z q z. ;; This proof requires one to recall the cosine rule of triangles. A simpler proof, that a student can easily derive, is given in the coming pages.
Bilinear property: For any vectors p, q, r in vector space v, and lambda in real numbers ;; lambda times vector p + vector q, dot vector r ; equals ; lambda times vector p dot vector r + vector q dot vector r. ;; This is explained and proven in properties of the dot product. For now, consider this to be true.
A vector p = p x i + p y j + p z k ; is sum of scalar multiple of vectors. i, j, k are unit vectors and the scalar multiples are p x, p y, p z. The same applies to vector q = q x i + q y j + q z k, sum of multiple of vectors.
Proof for component form of vector dot product using bilinear property of dot product.
For given two vectors in component forms, the dot product is the sum of product of corresponding components of the vectors.
Vector Dot Product in Component Form: For given two vectors vector p = p x i + p y j + p z k and vector q = q x i + q y j + q z k; vector p dot vector q = p x q x, + p y q y, + p z q z.
Given vector p and vector q, what is vector p dot vector q?
1
2
3
4
The answer is "minus point 2"

we are not perfect yet...