Parametric Functions, Vectors, and Matrices — Quiz 2

Questions in this quiz (9)

1. If a particle moves in the $$xy$$-plane so that at time $$t$$>$$0$$ its velocity vector is $$\vec{v}(t)=\langle t^2+1,\ 2t+3\rangle$$, what is the speed of the particle at $$t=2$$?

   • $$\sqrt{29}$$
   • $$\sqrt{34}$$
   • $$\sqrt{74}$$
   • $$\sqrt{45}$$

2. Find the value of $$k$$ such that $$\vec{a}=4\mathbf{i}+k\mathbf{j}$$ and $$\vec{b}=k\mathbf{i}-2\mathbf{j}$$ are perpendicular.

   • $$-2$$
   • $$0$$
   • $$2$$
   • $$4$$

3. Given a particle whose velocity vector is $$\vec{v}(t)=\langle 2t-1,\ t+3\rangle$$ in meters per minute when $$t$$>$$0$$, find the time $$t$$ when the speed of the particle is $$\sqrt{13}$$ meters per minute.

   • $$\frac{1}{2}$$
   • $$\frac{3}{5}$$
   • $$1$$
   • $$2$$

4. A particle moves on a plane curve so that at any time $$t$$>$$0$$ its $$x$$-coordinate is $$t^3-2t$$ and its $$y$$-coordinate is $$(3t+1)^2$$. Its velocity vector is $$\vec{v}(t)=\langle 3t^2-2,\ 6(3t+1)\rangle$$. What is the speed of the particle, to the nearest tenth, when $$t=1$$?

   • $$9.8$$
   • $$12.0$$
   • $$24.0$$
   • $$25.0$$

5. Find the value of $$x$$ and $$y$$ such that $$\vec{u}=\vec{v}$$ given that $$\vec{u}=(2x+1)\mathbf{i}+(y-4)\mathbf{j}$$ and $$\vec{v}=5\mathbf{i}+(2x-3)\mathbf{j}$$.

   • $$x=2,\ y=3$$
   • $$x=2,\ y=5$$
   • $$x=1,\ y=3$$
   • $$x=3,\ y=2$$

6. Given the vectors below, which of the following are parallel to each other? I. $$\vec{a}=\langle 2,-6\rangle$$ II. $$\vec{b}=\langle 1,-3\rangle$$ III. $$\vec{c}=2\mathbf{i}+6\mathbf{j}$$

   • I and II only
   • II and III only
   • I and III only
   • I, II, and III

7. Given the vectors below, which of the following would form an obtuse angle between the two vectors? I. $$\vec{u}=\langle 3,-1\rangle$$ and $$\vec{v}=\langle -2,4\rangle$$ II. $$\vec{s}=\mathbf{i}+2\mathbf{j}$$ and $$\vec{t}=2\mathbf{i}-4\mathbf{j}$$ III. $$\vec{a}=\langle 1,2\rangle$$ and $$\vec{b}=\langle 2,-1\rangle$$

   • I only
   • II only
   • I and II only
   • None of the above

8. Given $$|\vec{p}|=3$$, $$|\vec{q}|=4$$, and the angle between the two vectors is $$\theta=60^\circ$$, find $$\vec{p}\cdot\vec{q}$$.

   • $$6$$
   • $$9$$
   • $$12$$
   • $$18$$

9. Which of the following is a vector that has the opposite direction to $$\langle 4,-3\rangle$$ and has a length of $$10$$?

   • $$\langle -8,6\rangle$$
   • $$\langle -6,8\rangle$$
   • $$\langle 6,-8\rangle$$
   • $$\langle -10,0\rangle$$