Vector Projection : Online Physics Tutors : Tutorpace

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

Therefore the projection of vector a^à on vector b^à = (44/25, 33/25)

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

Therefore the projection of vector x^à on vector y^à = (-21/13, 14/13)


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Vector Projection

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