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3.3: Arithmetic on Vectors in 3-Dimensional Space

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  2. Last updated
    May 8, 2023
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Addition and Subtraction of Vectors

To add or subtract two vectors, add or subtract their corresponding components.

Example 3.3.1

Add the vectors →u=⟨2, 5, 4⟩ and →v=⟨4,2,1⟩,

Solution

To add the vectors →u=⟨2, 5, 4⟩ and →v=⟨4,2,1⟩, add their corresponding components.

→u + →v=⟨2+4,5+2, 4+1⟩= ⟨6,7, 5⟩

So, →u + →v=⟨6,7, 5⟩

A diagram showing two vectors in a 3-dimensional coordinate system. In the xyz plane, we have two vectors. We have vector u that has a starting point of (0,0,0) and a terminal point of (2,5,4). We have vector v that has a starting point of (0,0,0) and a terminal point of (4,2,1). There are lines that connect these vectors to the x-axis, the y-axis, and the z-axis.

Now, graph this sum. Start at the origin.

Since the x−component is 6, move 6 units in the x−direction.

Since the y−component is 7, move 7 units in the y−direction.

Since the z−component is 5, move 5 units upward.

A diagram showing two vectors in a 3-dimensional coordinate system and the graph of their sum. In the xyz plane, we have three vectors. We have vector u that has a starting point of (0,0,0) and a terminal point of (2,5,4). We have vector v that has a starting point of (0,0,0) and a terminal point of (4,2,1). Then, we have a vector u + v has a starting point of (0,0,0) and the terminal point is (2,5,1). There are lines that connect these vectors to the x-axis, the y-axis, and the z-axis.

Example 3.3.2

Subtract the vectors →u=⟨2, 5, 4⟩ and →v=⟨4,2, 1⟩

Solution

To subtract the vectors →u=⟨2, 5, 4⟩ and →v=⟨4,2, 1⟩ subtract their corresponding components.

→u− →v=⟨2−4,5−2, 4−1⟩= ⟨−2,3, 3⟩

So, →u− →v=⟨−2,3, 3⟩

Scalar Multiplication

Definition: Term

Scalar multiplication is the multiplication of a vector by a real number (a scalar).

Suppose we let the letter k represent a real number and →v be the vector ⟨x,y,z⟩. Then, the scalar multiple of the vector →v is

k→v=⟨kx,ky,kz⟩

Example 3.3.3

Suppose →u=⟨−3,−8, 5⟩ and k=3.

Solution

Then k→u= 3→u=3⟨−3,−8, 5⟩= ⟨3(−3),3(−8),3(???)⟩=⟨−9,−24, 15⟩

Example 3.3.4

Suppose →v=⟨6,3,−12⟩ and k=−13.

Solution

Then k→u= −13→u=−13⟨6,3, −12⟩= ⟨−13(6),−13(3),−13(−12)⟩=⟨−2,−1, 4⟩

Example 3.3.5

Suppose →u=[−260], →v=[12−8], and →w=[−3−12]. Find 3→u+4→v−2→w.

Solution

Then 3→u+4→v−2→w=3[−260]+4[12−8]−2[−3−12]=[−6180]+[48−32]+[62−4]=[428−36]

Using Technology

We can use technology to determine the value of adding or subtracting vectors.

Go to www.wolframalpha.com.

Suppose →u=[−260], →v=[12−8], and →w=[−3−12]. Use WolframAlpha to find 3→u+4→v−2→w. In the entry field enter evaluate 3∗[−2,6,0]+4∗[1,2,−8]−2∗[−3,−1,2].

This screenshot from WolframAlpha shows an equation with scalar multiplication and vector addition and subtraction. We have 3 times <negative 2,6,0 plus 4 times <1,2, negative 8> minus 2 times . The result is <4,28, negative 36>." src="/@api/deki/files/97440/clipboard_e674088772cf81d9ce229ad6440ed593c.png">

WolframAlpha answers (4, 28, −36) which is WolframAlpha’s notation for [428−36].

Try These

Exercise 3.3.1

Add the vectors →u=⟨−3,4,6⟩ and →v=⟨8,7,−5⟩.

Answer

→u + →v=⟨5,11, 1⟩

Exercise 3.3.2

Subtract the vector →v=⟨8,7,−5⟩ from the vector →u=⟨−3,4,6⟩.

Answer

→u− →v=⟨−11,−3, 11⟩

Exercise 3.3.3

Given the three vectors, →u=⟨2,4,−5⟩,→v=⟨−3,4,−8⟩, and →w=⟨0,1,2⟩, find 2→u+3→v−4→w.

Answer

2→u+3→v−4→w =⟨−5,16, −42⟩

Exercise 3.3.4

Suppose →u=[34−2],→v=[−164], and →w=[052], find 4→u−4→v−→w

Answer

4→u−4→v−→w=⟨16,−13, −26⟩

This page titled 3.3: Arithmetic on Vectors in 3-Dimensional Space is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Denny Burzynski (Downey Unified School District) .

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