Two arrows are drawn in 3D space with identical components ⟨3, 0, −2⟩ — one with its tail at (1, 1, 0) and one with its tail at (5, −2, 3). Which statement is correct?
AThey are different vectors because they start at different locations in space
BThey are the same vector because vectors are defined by their components, not their starting position
CThey are different vectors because their heads point to different terminal points
DThey are equal only if one of them is a position vector from the origin
A vector is defined entirely by its components — its direction and magnitude. Two vectors with identical components are the same mathematical object regardless of where their tails are placed. This is what distinguishes vectors from points: vectors are free to be translated anywhere without changing their identity. Option A is the classic misconception of treating a vector like a fixed, located arrow. Option C describes points (locations), not vectors.
Question 2 Multiple Choice
A particle is at point P = (2, −1, 4). You apply displacement vector v = ⟨−2, 3, −4⟩. What is the particle's new position?
C(0, 2, 0) — adding v's components to P's coordinates
D(4, −4, 8) — subtracting the wrong way
Applying a displacement vector means adding its components to the current coordinates: (2 + (−2), −1 + 3, 4 + (−4)) = (0, 2, 0). Option A is the most common error — confusing the vector (a displacement instruction) with the resulting position. The vector tells you how far to move; the new position is where you end up after starting at P.
Question 3 True / False
The magnitude of vector ⟨3, 0, 4⟩ is 5.
TTrue
FFalse
Answer: True
The magnitude is computed using the 3D Pythagorean theorem: ‖v‖ = √(3² + 0² + 4²) = √(9 + 0 + 16) = √25 = 5. The zero y-component simply contributes nothing under the radical — this vector lies in the xz-plane and is the familiar 3-4-5 right triangle extended into 3D.
Question 4 True / False
The position vector of point (a, b, c) is a fundamentally different type of mathematical object from the vector ⟨a, b, c⟩.
TTrue
FFalse
Answer: False
A position vector IS the vector ⟨a, b, c⟩, conventionally drawn with its tail at the origin. It is not a new type of object — it is the same displacement vector, just given a specific starting point. The term 'position vector' describes a role or convention (tail at origin), not a different mathematical entity. This bridge between points and vectors is exactly what makes the notation r = ⟨x, y, z⟩ useful for tying geometry to algebra.
Question 5 Short Answer
What is the key conceptual difference between a point (x, y, z) and a vector ⟨x, y, z⟩, and how does the notion of a position vector bridge the two?
Think about your answer, then reveal below.
Model answer: A point (x, y, z) is a fixed location in space. A vector ⟨x, y, z⟩ is a displacement — an instruction to move x units along the x-axis, y along y, z along z — with no fixed starting position. The position vector bridges them by anchoring the vector's tail at the origin, so it points from (0,0,0) to the point (x,y,z). This lets every point be identified with the displacement needed to reach it from the origin.
This distinction is foundational. Vectors can be freely translated (same vector, different starting point); points cannot. The confusion arises because they share numerical coordinates — both a point and a vector can be described by three numbers — but they answer different questions: 'where is it?' vs. 'how far and in what direction?' The position vector is useful precisely because it gives each point a canonical vector representative, linking the geometry of locations to the algebra of displacements.