# Vector Projection

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A vector is generally displayed as a line with an arrow and a vector is used to represent magnitude or direction of a particular physical quantity. Vectors can be drawn on the coordinate axis as well, depending upon the location of a certain quantity. On a 2-dimensional X-Y coordinate plane or a 3-dimensional coordinate plane, a vector can be drawn based on the given location. The projection of one vector on another given vector is known as vector projection and it can be calculated using the vector projection formula!

Example 1: Find the vector projection of the vector aà= (2, 1) on vector bà= (4, 3)?
Given: aà= (2, 1) and bà= (4, 3)
Vector projection formula, proj ba = [(aà. bà)/ |bà|2] bà
Applying the above formula, we first calculate: (aà. bà) = (2* 4) + (1* 3) = 8+ 3= 11
And now, | bà | = √(42 + 32) = √(25)= 5
So: | bà |2 =52 = 25
So according to the formula: proj ba= [11/25] (4, 3)

Therefore the projection of vector aà on vector bà = (44/25, 33/25)

Example 2: Find the vector projection of the vector xà= (1, 5) on vector yà= (-3, 2)?
Given: xà= (1, 5) and yà= (-3, 2)
Vector projection formula, proj yx = [(xà. yà)/ |yà|2] yà
Applying the above formula, we first calculate: (xà. yà) = (1* -3) + (5* 2) = -3+ 10= 7
And now, | yà | = √((-3)2 + 22) = √(13)
So: | yà |2 = (√13)2 = 13
So according to the formula: proj yx= [7/13] (-3, 2)

Therefore the projection of vector xà on vector yà = (-21/13, 14/13)