Math Archive: Questions from 2024-09-10

Which of the following expressions are meaningful? Which are meaningless? Explain. (a) (a⋅b)⋅c The expression (a⋅b)⋅c has --- Select--- because it is the dot product of -- Select---. (b) (a⋅b)c The expression (a⋅b)c has --- Select--- because it is a scalar multiple of --- Select---. (c) |a|(b⋅c) The expression |a|(b⋅c) has --- Select--- ∨ because it is the product of --- Select---. (d) a⋅(b+c) The expression a⋅(b+c) has --- Select--- because it is the dot product of --- Select---. (e) a⋅b+c The expression a⋅b+c has --- Select--- because it is the sum of --- Select---. (f) |a|⋅(b+c) The expression |a|⋅(b+c) has --- Select--- because it is the dot product of --- Select---.

Find a⋅b. a = ⟨5, −5, 4⟩, b = ⟨5, 6, −1⟩

Find a⋅b. a = 4i + 4j − k, b = −3i + 8k

Find a⋅b. |a| = 80, |b| = 90, the angle between a and b is 3π/4.

If u is a unit vector, find u⋅v and u⋅w. (Assume v and w are also unit vectors.) u⋅v = u⋅w =

If u is a unit vector, find u⋅v and u⋅w. (Assume w is also a unit vector.) u⋅v = u⋅w =

A street vendor sells a hamburgers, b hot dogs, and c bottles of water on a given day. He charges $5 for a hamburger, $4.50 for a hot dog, and $2 for a bottle of water. If A = ⟨a, b, c⟩ and P = ⟨5, 4.5, 2⟩, what is the meaning of the dot product A⋅P? The dot product is the amount the vendor receives when a customer buys all three items. The dot product is the average amount of revenue the vendor can expect on a given day. The dot product is equal to the vendor's total revenue for the day. The dot product gives the total number of items the vendor sells on that day.

Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) a = ⟨1, −4, 1⟩, b = ⟨0, 5, −5⟩ exact approximate

Determine whether the given vectors are orthogonal, parallel, or neither. (a) a = ⟨9, 6⟩, b = ⟨−4, 6⟩ orthogonal parallel neither (b) a = ⟨4, 7, −2⟩, b = ⟨3, −1, 7⟩ orthogonal parallel neither (c) a = −4i + 12j + 8k, b = 3i − 9j − 6k orthogonal parallel neither (d) a = 2i − j + 2k, b = 3i + 2j − 2k orthogonal parallel neither

Consider the triangle with vertices P(0, −4, −3), Q(1, −1, −5), and R(5, −3, −6). Determine the following vectors. QP→ = QR→ = Find QP→⋅QR→ QP→⋅QR→ = Is the given triangle right-angled? Yes, it is right-angled. No, it is not right-angled.

Find the direction cosines and direction angles of the vector. (Give the direction angles correct to the nearest tenth of a degree.) ⟨−7, 3, 3⟩ cos⁡(α) = ∘ cos⁡(β) = ∘ cos⁡(γ) = ∘ α = ∘ β = ∘ γ = ∘

Find the scalar and vector projections of b onto a. a = ⟨6, 7, −6⟩ b = ⟨3, −1, 1⟩ scalar projection of b onto a vector projection of b onto a

Find the scalar and vector projections of b onto a. a = i + 4j + 5k, b = 9i − k scalar projection of b onto a vector projection of b onto a