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To find u + v , we first draw the vector u , and from the terminal end of u , we drawn the vector v . In other words, we have the initial point of v meet the terminal end of u . This position corresponds to the notion that we move along the first vector and then, from its terminal point, we move along the second vector. The sum u + v is the resultant vector because it results from addition or subtraction of two vectors. The resultant vector travels directly from the beginning of u to the end of v in a straight path, as shown in [link] .

Diagrams of vector addition and subtraction.

Vector subtraction is similar to vector addition. To find u v , view it as u + (− v ). Adding − v is reversing direction of v and adding it to the end of u . The new vector begins at the start of u and stops at the end point of − v . See [link] for a visual that compares vector addition and vector subtraction using parallelograms .

Showing vector addition and subtraction with parallelograms. For addition, the base is u, the side is v, the diagonal connecting the start of the base to the end of the side is u+v. For subtraction, thetop is u, the side is -v, and the diagonal connecting the start of the top to the end of the side is u-v.

Adding and subtracting vectors

Given u = 3 , 2 and v = −1 , 4 , find two new vectors u + v , and u v .

To find the sum of two vectors, we add the components. Thus,

u + v = 3 , 2 + 1 , 4 = 3 + ( 1 ) , 2 + 4 = 2 , 2

See [link] (a) .

To find the difference of two vectors, add the negative components of v to u . Thus,

u + ( v ) = 3 , 2 + 1 , 4 = 3 + 1 , 2 + ( 4 ) = 4 , 6

See [link] (b).

Further diagrams of vector addition and subtraction.
(a) Sum of two vectors (b) Difference of two vectors
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Multiplying by a scalar

While adding and subtracting vectors gives us a new vector with a different magnitude and direction, the process of multiplying a vector by a scalar    , a constant, changes only the magnitude of the vector or the length of the line. Scalar multiplication has no effect on the direction unless the scalar is negative, in which case the direction of the resulting vector is opposite the direction of the original vector.

Scalar multiplication

Scalar multiplication involves the product of a vector and a scalar. Each component of the vector is multiplied by the scalar. Thus, to multiply v = a , b by k , we have

k v = k a , k b

Only the magnitude changes, unless k is negative, and then the vector reverses direction.

Performing scalar multiplication

Given vector v = 3 , 1 , find 3 v , 1 2 v , and − v .

See [link] for a geometric interpretation. If v = 3 , 1 , then

3 v = 3 3 , 3 1 = 9 , 3 1 2 v = 1 2 3 , 1 2 1 = 3 2 , 1 2 v = −3 , −1
Showing the effect of scaling a vector: 3x, 1x, .5x, and -1x. The 3x is three times as long, the 1x stays the same, the .5x halves the length, and the -1x reverses the direction of the vector but keeps the length the same. The rest keep the same direction; only the magnitude changes.
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Find the scalar multiple 3 u given u = 5 , 4 .

3 u = 15 , 12

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Using vector addition and scalar multiplication to find a new vector

Given u = 3 , 2 and v = 1 , 4 , find a new vector w = 3 u + 2 v .

First, we must multiply each vector by the scalar.

3 u = 3 3 , 2 = 9 , 6 2 v = 2 1 , 4 = 2 , 8

Then, add the two together.

w = 3 u + 2 v = 9 , 6 + 2 , 8 = 9 2 , 6 + 8 = 7 , 2

So, w = 7 , 2 .

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Finding component form

In some applications involving vectors, it is helpful for us to be able to break a vector down into its components. Vectors are comprised of two components: the horizontal component is the x direction, and the vertical component is the y direction. For example, we can see in the graph in [link] that the position vector 2 , 3 comes from adding the vectors v 1 and v 2 . We have v 1 with initial point ( 0 , 0 ) and terminal point ( 2 , 0 ) .

Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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