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

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 = −5 2 + 6 2 =61

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.

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.

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=−2 𝑏=−1 𝑐=6 ∴𝑃(−2,−1,6).

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)