Mathematics IM

Worked Examples

ALGEBRA: VECTORS

Produced by the Maths Learning Centre, The University of Adelaide.

May 3, 2013

The questions on this page have worked solutions and links to videos on the followingpages. Click on the link with each question to go straight to the relevant page. You willneed to have the question handy to refer to while watching the videos.

Questions

1. See Page 3 for worked solutions.In each part below draw a sketch to illustrate the answer.

(a) Find the components of the vector with initial point (4, 5) and terminal point(−1,−5).

(b) Find the components of the vector with initial point (4,2) in the direction of thevector [−2, 6] which has terminal point (0, y) for some y ∈ R.

(c) Find the components of the vector with intial point (4,2) in the direction of [2, −6]which has terminal point (x, 0) for some x ∈ R.

2. See Page 4 for worked solutions.Are there numbers (scalars) a, b ∈ R such that a[1, 2] + b[−1, 1] = [8, 4]?

3. (a) See Page 5 for worked solutions.Can you find a vector u ∈ R2 such that u · u = 2 and u · [1, 1] = 0? Are thereany other such vectors?

(b) See Page 6 for worked solutions.Suppose that for u,v ∈ R3 we have ‖u‖ = 1 and ‖v‖ = 2. What are the largestand smallest possible values of ‖u− v‖?

4. See Page 7 for worked solutions.Find all unit vectors which are perpendicular to both a = [−1, −2, 2]and b = [1, 3, −1].

5. See Page 8 for worked solutions.Let u = [−1, 2, 3], v = [2, 1, −2] and w = [0, −3, 1] be vectors in R3. Findu · (v ×w), w · (u× v) and v · (w × u). What do you notice?

6. (a) See Page 10 for worked solutions.Find the equation of the line l through P (3, 2, 6) and Q(−1, 0, 4) in vector, para-metric and cartesian forms.

(b) See Page 11 for worked solutions.Find the equation of the line k which is perpendicular to l and passes through thepoints R(1, 1, 1) and X, a point on l.

7. (a) See Page 12 for worked solutions.Find the equation of the plane containing the points P (1, 1,−1), Q(3, 2,−6) andR(−1, 0,−2).

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(b) See Page 13 for worked solutions.Find the line through T (0, 2, 3) which is normal to this plane. Where does itintersect the plane?

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1. Click here to go to question list.In each part below draw a sketch to illustrate the answer.

(a) Find the components of the vector with initial point (4, 5) and terminal point(−1,−5).

(b) Find the components of the vector with initial point (4,2) in the direction of thevector [−2, 6] which has terminal point (0, y) for some y ∈ R.

(c) Find the components of the vector with intial point (4,2) in the direction of [2, −6]which has terminal point (x, 0) for some x ∈ R.

Click here to see video of this example on YouTube.

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2. Click here to go to question list.Are there numbers (scalars) a, b ∈ R such that a[1, 2] + b[−1, 1] = [8, 4]?

Click here to see video of this example on YouTube.

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3. (a) Click here to go to question list.Can you find a vector u ∈ R2 such that u ·u = 2 and u · [1, 1] = 0? Are there anyother such vectors?

Click here to see video of this example on YouTube.

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(b) Click here to go to question list.Suppose that for u,v ∈ R3 we have ‖u‖ = 1 and ‖v‖ = 2. What are the largestand smallest possible values of ‖u− v‖?Click here to see video of this example on YouTube.

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4. Click here to go to question list.Find all unit vectors which are perpendicular to both a = [−1, −2, 2] and b =[1, 3, −1].

Click here to see video of this example on YouTube.

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5. Click here to go to question list.Let u = [−1, 2, 3], v = [2, 1, −2] and w = [0, −3, 1] be vectors in R3. Findu · (v ×w), w · (u× v) and v · (w × u). What do you notice?

Click here to see video of this example on YouTube.

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6. (a) Click here to go to question list.Find the equation of the line l through P (3, 2, 6) and Q(−1, 0, 4) in vector, para-metric and cartesian forms.

Click here to see video of this example on YouTube.

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(b) Click here to go to question list.Find the equation of the line k which is perpendicular to l and passes through thepoints R(1, 1, 1) and X, a point on l.

Click here to see video of this example on YouTube.

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7. (a) Click here to go to question list.Find the equation of the plane containing the points P (1, 1,−1), Q(3, 2,−6) andR(−1, 0,−2).

Click here to see video of this example on YouTube.

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(b) Click here to go to question list.Find the line through T (0, 2, 3) which is normal to this plane. Where does itintersect the plane?

Click here to see video of this example on YouTube.

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