Position, Displacement, and Velocity Vectors
In two dimensions, direction lives in an angle and magnitude lives in a length. This quiz tests your command of the three central vector quantities of plane kinematics: position, displacement, and velocity.
You will be asked to:
- Write and interpret a position vector ##\vec{r} = x\hat{i} + y\hat{j}## and find its magnitude and direction.
- Compute displacement as a difference of position vectors.
- Distinguish average velocity from instantaneous velocity, and speed from velocity.
- Resolve velocity into components and reason about units and direction.
Keep the chain of definitions in mind: position is a vector, displacement is a difference of vectors, and velocity is a displacement divided by a time interval. Work carefully with signs and quadrant corrections.
Q1. A position vector is best described as a vector drawn from:
The position vector locates a particle relative to a chosen origin, so it runs from the origin to the particle and encodes both distance and direction.
Q2. A particle sits at coordinates ##(x, y)##. Its position vector is:
With unit vectors ##\hat{i}## and ##\hat{j}## along the axes, the position vector is ##\vec{r} = x\hat{i} + y\hat{j}##, where the components are signed numbers.
Q3. The magnitude of the position vector ##\vec{r} = x\hat{i} + y\hat{j}## equals:
The magnitude ##|\vec{r}| = \sqrt{x^2+y^2}## is the straight-line distance from the origin to the particle.
Q4. A particle is at ##(-3, -4)##. What is the magnitude of its position vector?
##|\vec{r}| = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9+16} = \sqrt{25} = 5##. The signs of the components do not affect the magnitude.
Q5. Why is a quadrant correction needed when using ##\theta = \tan^{-1}(y/x)## for the direction of a position vector?
The inverse tangent returns values over a limited range, so it cannot separate directions that differ by ##180^\circ##; a quadrant correction fixes this when ##x## is negative.
Q6. Displacement between two instants is defined as:
Displacement is a difference of vectors: ##\Delta\vec{r} = \vec{r}_2 - \vec{r}_1##, pointing from the initial to the final position.
Q7. A particle moves from ##\vec{r}_1 = 2\hat{i} + 3\hat{j}## to ##\vec{r}_2 = 5\hat{i} + 7\hat{j}##. Its displacement is:
##\Delta\vec{r} = \vec{r}_2 - \vec{r}_1 = (5-2)\hat{i} + (7-3)\hat{j} = 3\hat{i} + 4\hat{j}##.
Q8. A particle moves from ##\vec{r}_1 = 4\hat{i} - 2\hat{j}## to ##\vec{r}_2 = -1\hat{i} + 6\hat{j}##. Its displacement is:
##\Delta\vec{r} = (-1-4)\hat{i} + (6-(-2))\hat{j} = -5\hat{i} + 8\hat{j}##. Subtracting a negative component gives a positive result.
Q9. The magnitude of the displacement ##\Delta\vec{r} = 3\hat{i} + 4\hat{j}## is:
##|\Delta\vec{r}| = \sqrt{3^2 + 4^2} = \sqrt{9+16} = 5##.
Q10. Average velocity over a time interval ##\Delta t## is given by:
Average velocity is the displacement divided by the time interval, so it is a vector pointing along the displacement.
Q11. A particle has displacement ##6\hat{i} + 8\hat{j}## m in ##2## s. Its average velocity is:
##\vec{v}_{av} = \dfrac{\Delta\vec{r}}{\Delta t} = \dfrac{6\hat{i}+8\hat{j}}{2} = 3\hat{i} + 4\hat{j}## m/s. Dividing a vector by a scalar scales both components.
Q12. Instantaneous velocity is best defined as:
Instantaneous velocity is the limiting value of ##\Delta\vec{r}/\Delta t## as ##\Delta t \to 0##, giving the velocity at a single instant.
Q13. Which statement correctly distinguishes speed from velocity?
Speed is a scalar equal to the magnitude of the velocity vector; velocity carries both magnitude and direction.
Q14. A velocity vector is ##\vec{v} = 3\hat{i} + 4\hat{j}## m/s. What is the speed of the particle?
Speed is the magnitude of velocity: ##|\vec{v}| = \sqrt{3^2+4^2} = 5## m/s.
Q15. The SI unit of the position vector and of the displacement vector is:
Position and displacement are lengths, so both are measured in metres; velocity is measured in ##\text{m/s}##.
Q16. The SI unit of average velocity is:
Velocity is displacement divided by time, so its unit is metres per second, ##\text{m/s}##.
Q17. The direction of the average velocity vector over an interval is:
Since ##\vec{v}_{av} = \Delta\vec{r}/\Delta t## and ##\Delta t > 0##, the average velocity points in the same direction as the displacement.
Q18. A particle moves on a circle of fixed radius about the origin. As it moves, the magnitude of its position vector:
The position vector's magnitude equals the fixed radius, so it stays constant. As the particle moves around the circle, the vector's direction changes.
Q19. A drone moves from ##\vec{r}_1 = 1\hat{i} + 2\hat{j}## m to ##\vec{r}_2 = 4\hat{i} + 6\hat{j}## m in ##3## s. Its average velocity is:
##\Delta\vec{r} = 3\hat{i} + 4\hat{j}## m, so ##\vec{v}_{av} = \dfrac{3\hat{i}+4\hat{j}}{3} = 1\hat{i} + \dfrac{4}{3}\hat{j}## m/s.
Q20. A particle's position vector is written as ##\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}##. As the particle moves, the tip of this vector:
At each time, the tip of the position vector gives the particle's position. As time varies, that tip traces the particle's trajectory in the plane.