Eng Math 1, QB1 - Vectors

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

1a) What is 𝑎+𝑏?

Evaluate responses using AI:

OFF

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

1b) What is 𝑎 ∙ 𝑐?

Evaluate responses using AI:

OFF

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

1c) What is 𝑐×𝑏

Evaluate responses using AI:

OFF

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

1d) Harder: What is 𝑎 ×(𝑏×𝑐)

Evaluate responses using AI:

OFF

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?

Evaluate responses using AI:

OFF

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)

Evaluate responses using AI:

OFF

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)

Evaluate responses using AI:

OFF

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.

Evaluate responses using AI:

OFF

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

Evaluate responses using AI:

OFF

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?

Evaluate responses using AI:

OFF