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QUESTION: 1

If are any two vectors, then

Solution:

|a+b| ≤ |a| + |b|

Let us take an example : a = 1, b= 2

|1 + 2| ≤ |1| + |2|

|3| ≤ |3|

Hence, proved

QUESTION: 2

The angle between the vectors is: is :

Solution:

a = 6i - 3j + 2k b = 2i + j - 2k

a.b = 12 - 3 - 4 = 5

|a| = [(6)^{2} + (-3)^{2} + (2)^{2}]^{1/2 }

|a| = [36 + 9 + 4]^{½}

|a| = (49)^{½}

|a| = 7

|b| = [(2)^{2} + (1)^{2} + (-2)^{2}]^{½}

|b| = [4 + 1 + 4]^{½}

|b| = 3

Cosθ = (a.b)/|a||b|

= 5/(7)(3)

= 5/21

θ = cos^{-1}(5/21)

QUESTION: 3

If are two vectors, such that , then = ……

Solution:

|a - b|^{2} = |a|^{2} + |b|^{2} - 2|a||b|

|a - b|^{2} = (3)^{2} + (2)^{2} - 2(5)

|a - b|^{2} = 9 + 4 - 10

|a - b|^{2} = 3

|a - b| = (3)^{½.}

QUESTION: 4

The projection of the vector on the vector is:

Solution:

Projection = (A.B)/|B|

= [(i + 2j + k) . (2i + 3j + 2k)]/[(2)^{2} + (3)^{2} + (2)^{2}]^{½}

= (2 + 6 + 2)/[4 + 9 + 4]^{½}

= 10/(17)^{1/2}

QUESTION: 5

The angle between two non-zero vectors is given by

Solution:

A sequence is a function whose domain is the set of natural numbers or a subset of the natural numbers. We usually use the symbol an to represent a sequence, where n is a natural number and an is the value of the function on n. A sequence may be finite or infinite.

### Section Formula and Scalar Product of Two Vectors

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