𝑎 = 5𝑖̂ − 2𝑗̂ + 3𝑘̂ 𝑏 = −7𝑖̂+ 5𝑘̂ 𝑐 = 4𝑗̂ − 𝑘̂

1b) What is 𝑎 ∙ 𝑐?

𝑎 = 5𝑖̂ − 2𝑗̂ + 3𝑘̂ 𝑏 = −7𝑖̂+ 5𝑘̂ 𝑐 = 4𝑗̂ − 𝑘̂

1c) What is 𝑐×𝑏

𝑎 = 5𝑖̂ − 2𝑗̂ + 3𝑘̂ 𝑏 = −7𝑖̂+ 5𝑘̂ 𝑐 = 4𝑗̂ − 𝑘̂

1d) Harder: What is 𝑎 ×(𝑏×𝑐)

2 Two tugboats are towing a cargo ship. Tugboat A exerts a force of 15,000 N at a 30° angle North of East, while tugboat B exerts a force of 20,000 N with direction unknown. At what angle must tugboat B exert force for the cargo ship to travel in a straight line?

3a) If Vector 𝑢 = 5𝑖̂ + 4𝑗̂ − 2𝑘̂, and vector 𝑣 = −2𝑖̂ + 3𝑗̂ + 𝑘̂, determine if u and v are perpendicular (Hint, use the dot product)

3b) If Vector 𝑢 = −2𝑖̂ − 3𝑗̂ + 2𝑘̂, and vector 𝑣 = −4𝑖̂ + 2𝑗̂ +3𝑘̂, find the angle between u and v in both degrees and radians (Hint, use the dot product)

4a) A parallelepiped is represented by three vectors a, b & c, where:

𝑎 = 3𝑖̂ + 3𝑗̂ + 4𝑘̂, 𝑏 = −2𝑖̂ + 2𝑗̂ −1𝑘̂ and 𝑐 = −3𝑖̂ + 2𝑗̂ − 7𝑘̂.

Using these vectors, find the volume of the parallelepiped.

4b) If the volume of a parallelepiped is 26 cubic units, with vectors:

𝑎 = 2𝑖̂ + 𝑥𝑗̂ + 3𝑘̂, 𝑏 = −3𝑖̂+4𝑘̂ & 𝑐 = 5𝑖̂ −2𝑗̂ −4𝑘̂, find the value of 𝑥

5) A wrench is described by vector 𝑟̃ (given in metres m). Force 𝐹̃ (given in Newtons N) is applied.

a) If 𝑟̃=6𝑖̂+8𝑗̂ and 𝐹̃=4𝑖̂+2𝑗̂+2𝑘̂, what is the torque vector on the wrench? (Give Units)

b) How can the torque be maximised?

c) What is magnitude of the maximised torque if the magnitudes of vectors 𝑟̃ and 𝐹̃ remain constant?

6) If 𝑧 = −2 – 7𝑖 and 𝑤 = 3 + 4𝑖, find:

a) |𝑧|

b) 𝑎𝑟𝑔[𝑧]

6) If 𝑧 = −2 – 7𝑖 and 𝑤 = 3 + 4𝑖, find:

c) 𝑧 + 3𝑤

6) If 𝑧 = −2 – 7𝑖 and 𝑤 = 3 + 4𝑖, find:

d) 𝑧𝑤

6) If 𝑧 = −2 – 7𝑖 and 𝑤 = 3 + 4𝑖, find:

e) 𝑧/𝑤

f) What quadrant is 𝑧/𝑤 in?

7) If 𝑧 = 3𝑒^(𝑖5𝜋/4) and 𝑤 = 2𝑒^(−𝑖3𝜋/2) , find:

a) 𝑧𝑤

7) If 𝑧 = 3𝑒^(𝑖5𝜋/4) and 𝑤 = 2𝑒^(−𝑖3𝜋/2) , find:

b) 𝑧/𝑤

7) If 𝑧 = 3𝑒^(𝑖5𝜋/4) and 𝑤 = 2𝑒^(−𝑖3𝜋/2) , find:

c) 𝑤 in cartesian form

8) If 𝑧 = 𝑎 + 8𝑖 and 𝑤 = 3 – 𝑏𝑖 ,

find values for a and b such that 𝑧 – 2𝑤 = 0

9) Find the cube roots of 2 – 2𝑖

10) Find two solutions to the quadratic 2𝑥^2 − 5𝑥 + 5 = 0