1. 2.2 Operations on Algebraic Vectors
Consider the two vectors, 𝑢 = 𝑎,𝑏,𝑐 and 𝑣 =[𝑥,𝑦,𝑧] and the scalar 𝑘∈ℝ.
Vector Addition
𝑢 + 𝑣 = 𝑎+𝑥,𝑏+𝑦,𝑐+𝑧
Proof of Vector Addition:
𝑢 + 𝑣
=𝑎 𝑖 +𝑏 𝑗 +𝑐 𝑘 +𝑥 𝑖 +𝑦 𝑗 +𝑧 𝑘
= 𝑎+𝑥 𝑖 + 𝑏+𝑦 𝑗 +(𝑐+𝑧) 𝑘
= 𝑎+𝑥,𝑏+𝑦,𝑐+𝑧
Scalar Multiplication
𝑘 𝑢 = 𝑘𝑎,𝑘𝑏,𝑘𝑐

2. Example 1: If 𝑢 = 2,3,−1 and 𝑣 =[0,1,4] find 𝑤 =5 𝑢 −2 𝑣 .
Solution:
𝑤 =5 𝑢 −2 𝑣
=5 2,3,−1 −2[0,1,4]
= 10,15,−5 − 0,2,8
= 10,13,−13
*So basically just like “regular” algebra!
Example 2: Given 𝑎 =3 𝑖 −3 𝑗 + 𝑘 and 𝑏 = 𝑖 +2 𝑘 , calculate −2 𝑎 + 𝑏 .
Solution:
−2 𝑎 + 𝑏
=−2 3 𝑖 −3 𝑗 + 𝑘 + 𝑖 +2 𝑘
=−6 𝑖 +6 𝑗 −2 𝑘 + 𝑖 +2 𝑘
=−5 𝑖 +6 𝑗
∴ −2 𝑎 + 𝑏 = − =61

3. Example 3: Given points 𝐴(1,−4) and 𝐵 3,−1 , find 𝐴𝐵 .
Solution:
We can form the two position vectors 𝑂𝐴 =[1,−4] and 𝑂𝐵 =[3,−1] then,
𝐴𝐵 = 𝐴𝑂 + 𝑂𝐵
=− 𝑂𝐴 + 𝑂𝐵
= −1,4 +[3,−1]
=[2,3] * [3-1, -1-(-4)] 𝑂 𝐵 𝐴
In general when given two points 𝐴 𝑎 𝑥 , 𝑎 𝑦 and B 𝑏 𝑥 , 𝑏 𝑦 then 𝐴𝐵 = 𝑏 𝑦 − 𝑏 𝑥 , 𝑎 𝑦 − 𝑎 𝑥 i.e. the distance between the two points.

4. Example 4:
Using vectors, show that the points 𝑃 −1,−3 , 𝑄 0,2 and 𝑅(3,17) are collinear.
Solution:
𝑃𝑄 = 0− −1 ,2− −3 =[1,5] and 𝑄𝑅 = 3−0,17−2 =[3,15]
Recall that two vectors, 𝑎 and 𝑏 are parallel iff there exists a 𝑘∈ℝ such that 𝑎 =𝑘 𝑏 .
𝑃𝑄 =3 𝑄𝑅 ⇒ 𝑃𝑄 𝑄𝑅
∴𝑃, 𝑄, 𝑎𝑛𝑑 𝑅 are collinear.

5. Example 5:
If 𝐴 1,−5,2 and 𝐵(−3,4,4) are opposite vertices of a parallelogram 𝑂𝐴𝑃𝐵, with 𝑂 at the origin, find the coordinates of 𝑃.
Solution:
Let 𝑃 𝑎,𝑏,𝑐 .
𝐵𝑃 = 𝑂𝐴 = [1,−5,2] and 𝐴𝑃 = 𝑂𝐵 = −3,4,4
𝐴𝑃 = 𝑎−1, 𝑏+5, 𝑐−2 =[−3,4,4]
𝑎−1=−3 𝑏+5=4 𝑐−2=4
𝑎=− 𝑏=− 𝑐=6
∴𝑃(−2,−1,6).

6. Example 6:
Find the point on the 𝑧−axis that is equidistant from the points 𝐴(−5,2,1) and 𝐵(2,−6,3).
Solution:
Let 𝑃 be the point on the 𝑧−axis that is equidistant form the points 𝐴(−5,2,1) and 𝐵(2,−6,3), then the 𝑥 and 𝑦 coordinates are 0 so 𝑃 is 0,0,𝑧 .
𝑃𝐴 =(−5,2,1−𝑧) and 𝑃𝐵 =(2,−6,3−𝑧)