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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.
mathsVector AlgebraMultiplication of Vectors by Scalar

### Vector as sum of vectors

In this page, you will learn that Vectors can be represented as sum of several vectors.

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Which of the following is the largest number?

• 5
• 2+3
• 6-1
• All the above
• All the above

A quantity can be equivalently represented with a number or combination of several numbers.

Three persons, starting from a common point, are walking
•  Person A walks at an angle 45^@ for 3xx sqrt2 meter
•  Person B walks 3 meter at an angle 0^@ and then 3 meter at an angle 90^@
•  Person C walks 6 meter at an angle 90^@ and then 3 xx sqrt2 meter at an angle −45^@

To walk to the starting point, all three has to walk in which of the following direction?

• same distance but different directions
• same distance in the same direction
• same distance in the same direction
• different distances in the same direction
• different distances in different directions

Answer is 'same distance in the same direction'. A vector quantity can be equivalently represented by combination of several vectors.

Three persons, starting from a common point, are walking
•  Person A walks at an angle 45^@ for 3xx sqrt2 meter
•  Person B walks 3 meter at an angle 0^@ and then 3 meter at an angle 90^@
•  Person C walks 6 meter at an angle 90^@ and then 3 xx sqrt2 meter at an angle −45^@. What describes the walk of Person A?

• One vector quantity 3sqrt2 at angle 45^@
• One vector quantity 3sqrt2 at angle 45^@
• Sum of two vector quantities 3sqrt2 and angle 45^@

The answer is 'One vector quantity'.

Three persons, starting from a common point, are walking
•  Person A walks at an angle 45^@ for 3xx sqrt2 meter
•  Person B walks 3 meter at an angle 0^@ and then 3 meter at an angle 90^@
•  Person C walks 6 meter at an angle 90^@ and then 3 xx sqrt2 meter at an angle −45^@ What describes the walk of Person B?

• two vector quantities 3 at 0^@ and 3 at 90^@
• two vector quantities 3 at 0^@ and 3 at 90^@
• two vector quantities 6 at 90^@ and 3sqrt2 at -45^@

Answer is 'two vector quantities 3 at 0^@ and 3 at 90^@'.

Note that person A’s and person B’s walks are specified by one vector quantity and two vector quantities. Still, they are at the same distance and direction from the starting point. Effectively, their return to the starting point can be specified by one identical vector quantity.

In scalar quantities, person A has 3 apples in a basket. Person B has 2 apples in one hand and 1 apple in another hand. Who has more apples?

• Person A
• Person B
• both have the same number of apples
• both have the same number of apples

The answer is 'both have the same number of apples', which is understood as 2+1 = 3.

Given vectors
vec x = 2i+3j+4k

vec y = -1i-1j-2k

vec z = i+2j+2k
What is vec x+vec y?

• i-2j+2k
• i+2j+2k
• i+2j+2k
• 3i+4j+6k
• none of the above

answer is 'i+2j+2k'. Note that the result is same as vec z. Reiterating that A vector can be equivalently given as a sum of vectors.

A vector can be equivalently given as the sum of vectors.

Linear Combination of Vectors: A vector can be equivalently represented as sum of vectors.

slide-show version coming soon