Vector Questions

Example 1

• r = [-1][1][-3] + λ [2][-2][5] . [5][a][b] lies on the line. Find a and b.
• [5][a][b] = [-1][1][-3] + λ [2][-2][5]. Now if we can work out λ then we can solve this equation
• 5 = -1 +2λ , 6=2λ, λ=3 and now we sub this into our other equations:
• a = 1-2λ = 1-2(3) = -5
• b = -3 + 5t = -3 + 5(3) = 12
• a=-5 and b=12

Example 2

• Show r = [-1][1][2] + λ [2][2][1] and r = [5][7][5] + t [6][6][3] represent the same line
• [6][6][3] = 3[2][2][1] meaning the direction vectors are multiples of each other, and therefore must be the same lines or parallel lines
• Sub in a staring point to the other equation ; [-1][1][2] = [5][7][5] + t [6][6][3]
• Equate x's ; -1 = 5 + 6t , -6=6t , t=-1
• Equate y's ; 1 = 7 +6t , -6=6t , t=-1
• Equate z's 2 = 5 + 3t , -3=3t , t=-1
• Therefore the 2 equations represent the same line

Example 3

• Show that r = [-1][1][2] + λ [2][2][1] is a different line to [-5][-3][1] + t [-4][-4][-2]
• Direction vectors are multiples of each other ; [-4][-4][-2] = -2 [2][2][1] so must be parallel or same line
• [-5][-3][1] = [-1][1][2] + λ [2][2][1]
• Equate x's ; -5 = -1 + 2λ , λ=3
• Equate y's ; -3 = 1 + 2λ , λ=-2
• Equate z's ; 1 = 2 + λ , λ=-1
• Therefore not the same line and must just be parallel

Example 4

• B(3,5,4) lies on r = [-1][1][2] + λ [2][2][1]. Find such positions of Q such that BQ = 15
• We can say Q = [-1][1][2] + λ [2][2][1] for a particular value of λ , and change this into a single vector: Q = [-1+2λ][1+2λ][2+λ]
• Now as we know BQ = OQ - OB , we can say ; [-1+2λ][1+2λ][2+λ] - [3][5][4] = [2λ-4][2λ-4][λ-2]
• Now we can do Pythagorous on this as we know |BQ|=15 , so: (2λ-4)^2 + (2λ-4)^2 + (λ-2)^2 = 15^2 . Expand to get 9λ^2 - 36λ - 189 = 0
• Simplify to get λ^2 - 4λ - 21 = 0. Factorise for (λ-7)(λ+3) and therefore λ = 7 or -3
• For both possbile positions of Q, sub in both 7 and -3 into Q = [-1+2λ][1+2λ][2+λ]
• 7 : [-1+2(7)][1+2(7)][2+(7)] = [13][15][9]
• -3 : [-1+2(-3)][1+2(-3)][2+(-3)] = [-7][-5][-1]
• Therefore the 2 possible points of Q are either [13][15][9] or [-7][-5][-1]