Practice Vectors in space
A vector describes a direction and a magnitude, often through coordinate components.
Vectors help you model displacements and calculate positions in space.
Exercises with solutions
A vector describes a displacement and has both length and direction. You write it as a column with three coordinates. Vectors are added component by component and multiplied by a number, a scalar. The magnitude is |v| = √(x² + y² + z²).
Exercise 1
Add a = (2; 3; 1) and b = (4; −1; 5).
Show solution and working
(6; 2; 6)
Add component by component: 2 + 4 = 6, 3 + (−1) = 2 and 1 + 5 = 6.
Exercise 2
Calculate 3 · (1; −2; 4).
Show solution and working
(3; −6; 12)
Each coordinate is multiplied by 3.
Exercise 3
Calculate the magnitude of v = (3; 4; 0).
Show solution and working
5
|v| = √(3² + 4² + 0²) = √(9 + 16) = √25 = 5.
Exercise 4
Are a = (2; 4; 6) and b = (1; 2; 3) parallel?
Show solution and working
Yes, because a = 2 · b
Two vectors are parallel when one is a multiple of the other. Here a is twice b.
Common mistake
Confusing vectors with points. A point has a fixed position, a vector only length and direction; the same vector fits in many places. And forgetting a coordinate when computing the magnitude.
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