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# How to Find Magnitude and Direction without A Graph?

TopWe can characterize any vector quantity by just two chief features that are its magnitude (also called as size) and direction (basically known from the angle formed by the vector). The arrows are used to represent the directions of vectors. For instance, a vector can have directions as: North, East, South, West, North East, South West, North West and South East. When the direction of vector changes, its magnitude remains the same but the angle it subtends with respect to referential axis i.e. x – axis changes. Let us see how to find magnitude and direction without a graph. When we do not have any graph for the vector, we use direct formula for evaluation of magnitude and direction of the vector. The formulae we use are as follows:

Suppose a vector is given as a I + b J + c K, where I, J and K represent two orthogonal unit cap (means their magnitude is 1) vectors, then the horizontal and vertical components of the vector are calculated as:

M = Magnitude = (a2 + b2 + c2)½

To find the direction with respect to x – axis:

Cos A = a / M

A = cos-1 a / M

To find the direction with respect to y – axis:

Cos A = b / M

A = cos-1 b / M

Horizontal component = magnitude (V) cos A

And Vertical component = magnitude (V) sin A

Let us consider an example of it:

Example: Suppose we have a vector 2 I + 2 J + 2 k, then find its magnitude and direction?

Solution: Magnitude = (22 + 22 + 22)½ = 2 31/2

Its direction with respect to x – axis will be:

Cos A = 2 / 2 31/2

A = cos-1 1 / 31/2