Final Exam

For any vectors u and v in V₃, and any scalar k, k(u . v) = (ku) . v

True

For any vectors u and v in V₃, and any scalar k, k(u x v) = (ku) x v

True

For any vectors u and v in V₃, u . v = v . u

True

For any vectors u and v in V₃, |u x v| = |v x u|

True

For any vectors u, v, and w in V₃, (u + v) x w = u x w + v x w

True

For any vectors u, v, and w in V₃, u . (v x w) = (u x v) . w

True

If F is a vector field, then curl F is a vector field

True

If u . v = 0 and u x v = 0, then u = 0 or v = 0

True

The derivative of a vector function is obtained by differentiating each component function

True

For any vectors u and v in V₃, (u + v) x v = u x v

True - (u + v) x v = (u x v) + (v x v) and (v x v) = 0

For any vectors u and v in V₃, (u x v) . u = 0

True - (u x v) is orthogonal to u

The curve r(t) = (0, t², 4t) is a parabola

True - x = 0, y = t², z = 4t so t = z/4 and y = 1/16z²

For any vectors u and v in V₃, |u + v| = |u| + |v|

False

For any vectors u and v in V₃, |u . v| = |u||v|

False

For any vectors u and v in V₃, |u x v| = |u||v|

False

If r(t) is a differentiable vector function, then d/dt[r(t)] = |r'(t)|

False

If F is a vector field, then div F is a vector field

False - div F is a scalar field

If u . v = 0, then u = 0 or v = 0

False - if u . v = 0, then the vectors are orthogonal

If u x v = 0, the u = 0 or v = 0

False - if u x v = 0, then the vectors are parallel

The set of points {(x, y, z) | x² + y² = 1} is a circle

False - in R³, this is a cylinder, not a circle

The curve r(t) = (2t, 3-t, 0) is a line that passes through the origin

False - there is not t-value that gives the point (0, 0, 0)

The vector (3, -1, 2) is parallel to the plane 6x - 2y + 4z = 1

False - these planes are proportional, but not parallel

A linear equation Ax + By + Cz + D = 0 represents a line in space

False - this is the equation of a plane

If u(t) and v(t) are differentiable vector functions, then d/dt[u(t) x v(t)] = u'(t) x v'(t)

False - this will equal u(t)v'(t) + v(t)u'(t) because of the product rule

If u = (u₁, u₂) and v = (v₁, v₂), then u . v = (u₁v₁, u₂v₂)

False - u . v is not a vector and u . v = u₁v₁ + u₂v₂ + ...

For any vectors u and v in V₃, u x v = v x u

False - u x v = -v x u