True or False Vector Calculus

A set of parametric equations for the line through (1, −2, 0) that is parallel to (−2, 4, 7) is x = 1 − 2t, y = 4t − 2, z = 7.

False

Displacement and distance are scalar quantities

False

For any vector a in Rn and scalar k, we have k||a|| = k||a|| .

False

For any vectors a, b, c in R3, we have a ×(b ×c) = (a ×b) ×c

False

If A = [1 2 0 3; -1 0 2 1; 5 9 2 0; 0 8 0 -6], then det A = 2[-1 2 1; 5 2 0; 0 0 -6] + 9 [1 0 3; -1 2 1; 0 0 -6] -8[1 0 3; -1 2 1; 5 2 0]

False

If A is a 5 × 7 matrix and B is a 7 × 7 matrix, then B A is a 7 × 5 matrix

False

If A is an n × n matrix, then det (2A) = 2 det A

False

If a = (1, 7, −9) and b = (1, −9, 7), then a = b.

False

If a particle is at the point (2, −1) in the plane and moves from that point with velocity vector v = (1, 3), then after 2 units of time have passed, the particle will be at the point (5, 1).

False

The displacement vector from P1(1, 0, −1) to P2(5, 3, 2) is (−4, −3, −3).

False

The dot product of two unit vectors is 1

False

The line with parametric equations x = 2 − 3t, y = t + 1, z = 2t − 3 has symmetric form (x + 2)/−3 = y − 1 = (z − 3)/2 .

False

The parametric equations x = 2 sin t, y = 2 cost, where 0 ≤ t ≤ π, describe a circle of radius 2.

False

The plane containing the points (1, 2, 1), (3, −1, 0), and (1, 0, 2) is given by the parametric equations x = 2s, y = −3s − 2t, z = t − s

False

The surface having equation ρ = 4 cos θ in spherical coordinates is a sphere of radius 2

False

The surface whose equation in cylindrical coordinates is z = 2r is the same as the surface whose equation in spherical coordinates is ϕ = π/6

False

Velocity and speed are vector quantities.

False

(a ×b)· c − (a ×c)· b is a scalar

True

A set of parametric equations for the line through (1, 2, 3) and (4, 3, 2) is x = 4 − 3t, y = 3 − t, z = t + 2.

True

Force and acceleration are vector quantities.

True

If a and b are two vectors in R3 and k and l are real numbers, then (k − l)(a + b) = ka − la + kb − lb.

True

If a, u ∈ R^n and u = 1, then proj(u)a = (a · u)u.

True

The plane containing the points (1, 2, 1), (3, −1, 0), and (1, 0, 2) has equation 5x + 2y + 4z = 13

True

The surface having equation r = 4 sin θ in cylindrical coordinates is a cylinder of radius 2.

True

The surface having equation ρ = 3 in spherical coordinates is the same as the surface whose equation in cylindrical coordinates is r^2 + z^2 = 9

True

The surface having equation ρ cos θ sin ϕ = 3 in spherical coordinates is a plane

True

The two sets of parametric equations x = 3t − 1, y = 2 − t, z = 2t + 5 and x = 2 − 6t, y = 2t + 1, z = 7 − 4t both represent the same line.

True

The vector (2, 3, −2) is the same as 2i + 3j − 2k.

True

The volume of a parallelepiped determined by the vectors a, b, c ∈ R3 is |(a ×c) · b|

True

||a||b − ||b||a is a vector

True