How to Solve a 3-d Vector Problem Using Dot Product?

A 3 – dimensional vector is formed by using three orthogonal unit vectors of three perpendicular axes of 3 – D planes. Unit vector for x – axis we generally use is “i”, for y – axis is “j” and for z – axis we use “k”. These unit vectors have magnitude equals to 1. For instance, 2 i + 3 j + 4 k represent a 3 – dimensional vector. How to solve a 3 d vector problem using dot product? In a dot product resulting quantity is also a vector quantity with a different magnitude. Application of dot product can be found in fields of sciences especially physics and also in maths. Suppose we have two vectors a i + b j + c k and h i + r j + s k, then their dot product can be evaluated as:

(a i + b j + c k) . (h i + r j + s k) = (a x h) i + (b x r) j + (c x s) k,

Dot product is similar to Scalar multiplication of two vectors which can be explained by means of an example as follows:
Suppose we have two vectors: 2 i + 3 j + 4 k and 2 i + 5 j + 7 k,

Dot product of these vectors can be evaluated as follows:
(2 i + 3 j + 4 k) . (2 i + 5 j + 7 k) = (2 x 2) i + (3 x 5) j + (4 x 7) k = 4 i + 15 j + 28 k,

Thus we see that resulting value is also a vector quantity whose magnitude can be calculated as follows:
R = √(4² + 15² + 28²),
Or R = √(16 + 225 + 784) = √(1025).