  If you like what you see in Mathguru
Subscribe Today
 For 12 Months US Dollars 12 / Indian Rupees 600 Available in 20 more currencies if you pay with PayPal. Buy Now No questions asked full moneyback guarantee within 7 days of purchase, in case of Visa and Mastercard payment WyzAnt Tutoring

Example:Finding Projection of a Vector

 Post to:   Explanation:

Dot product

In mathematics, the dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number obtained by multiplying corresponding entries and then summing those products. The name is derived from the centered dot "·" that is often used to designate this operation; the alternative name scalar product emphasizes the scalar (rather than vector) nature of the result. At a basic level, the dot product is used to obtain the cosine of the angle between two vectors.

## Geometric interpretation  is the scalar projection of onto .
Since , then .

In Euclidean geometry, the dot product of vectors expressed in an orthonormal basis is related to their length and angle. For such a vector , the dot product is the square of the length of , or where denotes the length (magnitude) of . If is another such vector, where θ is the angle between them.

This formula can be rearranged to determine the size of the angle between two nonzero vectors: (Our solved example in mathguru.com uses this concept)

http://en.wikipedia.org/wiki/Dot_product

# Vector projection

The vector projection (also known as the vector resolute, or vector component) of a vector in the direction of a vector (or "of on/onto "), is given by: where θ is the angle between the vectors and ; the operator is the dot product; and is the unit vector in the direction of .

(Our solved example in mathguru.com uses this concept)

The other component of (perpendicular to ), called the vector rejection of from , is given by: Both the vector projection and the vector rejection are vectors. The vector projection of on is the orthogonal projection of onto the straight line defined by . The corresponding vector rejection is the orthogonal projection of onto a plane orthogonal to .

The vector projection of on can be also regarded as the corresponding scalar projection multiplied by .

## Overview

If A and B are two vectors, the projection of A on B is the vector C with the same direction as B and with the length: When θ is not known, we can compute using the following property of the dot product : Thus, the length of C can be also computed as follows Since C is in the same direction as B, where is the unit vector with the same direction as B: Substituting, we obtain which is equivalent to either  The latter formula is computationally more efficient than the former, as the former requires three multiplications and a square root to compute |B| (in turn, needed to compute ), while the latter only requires three multiplications to compute (the remaining parts of the two formulas require the same number and kind of basic algebraic operations).

http://en.wikipedia.org/wiki/Vector_projection