# MCV4U_Unit#7_TCavers.pdf - MCV4U Lesson#18-20 u2013 Lines...

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MCV4U: Lesson #18-20 Lines and Planes (Part B) Taylor Cavers Student Number: 17144011
Lesson #20: Intersection of Lines and Planes in 3D-Space Task #1: Knowledge and Understanding Questions 1. Determine the angle between each of the following pairs of vectors. (10 marks: 5 marks each) . . .
b. ? ⃗ = (−?, ?, ?) ?𝒏𝒅 ? ⃗ = (?, −?, ?) Step #1: Calculate the dot product of ? ⃗ ∙ ? . ? ∙ ? = (−1, 4, 5) ∙ (3, −1, 3) ? ∙ ? = (−1)(3) + (4)(−1) + (5)(3) ? ∙ ? = −3 − 4 + 15 ? ∙ ? = 8 Step #2: Determine the magnitude of ? and ? . |?| = √(−1) 2 + (4) 2 + (5) 2 |?| = √1 + 16 + 25 |? | = √42 |? | = √(3) 2 + (−1) 2 + (3) 2 |? | = √9 + 1 + 9 |? | = √19 Step #3: Determine theta the angle between vectors ? ⃗⃗ and ? . cos(𝜃) = ? ∙ ? |?||? | cos 𝜃 = 8 (√42 )(√19 ) cos 𝜃 = 8 (√42 )(√19 ) cos 𝜃 = 8 √798 cos 𝜃 = 0. 2832 𝜃 = cos −1 (0.2832) 𝜃 = 73.5° 𝜃 = 74° Therefore, the angle between ? and ? is 74°.
2. Find the slope of the vector that is perpendicular to the scalar equation ?? − ?? + ? = ? . (2 marks)
3. Write an alternate vector equation for the following line. Change both the point and the direction vector: ? ⃗⃗⃗ = (?, −?, ?) + ?(−?, ?, ?) . (3 marks) .
4. Determine whether the angle between each of the following pairs of vectors is acute, obtuse, or neither. (4 marks: 2 marks each) . 2 .
5. Given the vector equation of a line in 2-space, (?, ?) = (?, ?) + ?(?, ?) , write a scalar equation for the line. (4 marks)