Topic 2 / 17basic

Vectors in the Plane and in Space

Represent vectors as arrows or components, compute magnitudes and unit vectors, add and scale them, and apply them to forces, tensions, and velocities.

A vector has both **magnitude** and **direction**. We represent vectors as arrows from a tail to a tip, or as ordered tuples of components. The same vector can sit anywhere in space — only its components matter.

Component form and i, j, k notation

A vector

v

in 3-space has the form

v = ⟨ v_{1}, v_{2}, v_{3} ⟩

, equivalent to

v_{1} i + v_{2} j + v_{3} k

, where

i, j, k

are the standard unit vectors along

x, y, z

. The notation

⟨ \cdot ⟩

distinguishes a vector from a point

(\cdot)

Key takeaways

$⟨ v_{1}, v_{2}, v_{3} ⟩ = v_{1} i + v_{2} j + v_{3} k$ .
Vectors are free: same components, same vector, regardless of base point.
Use $⟨ \cdot ⟩$ for vectors, $(\cdot)$ for points.

From two points: terminal minus initial

If a vector goes from

P (x_{1}, y_{1}, z_{1})

Q (x_{2}, y_{2}, z_{2})

, then

P Q = ⟨ x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1} ⟩

. **Terminal minus initial.** This is one of the most common sign-error sources in vector calculus.

Strategy tips

Always do tip minus tail: terminal point minus initial point.
Reversing direction reverses sign of every component.

Magnitude and unit vectors

The magnitude of

v = ⟨ v_{1}, v_{2}, v_{3} ⟩

∣ v ∣ = v_{1}^{2} + v_{2}^{2} + v_{3}^{2}

. A **unit vector** is a vector of length 1; the unit vector in the direction of

v

\hat{v} = v /∣ v ∣

. To construct, compute the magnitude first, then divide each component by it.

Key takeaways

Magnitude: square components, sum, take square root.
Unit vector $\hat{v} = v /∣ v ∣$ .
$∣ \hat{v} ∣ = 1$ always.

Sums, differences, scalar multiples

Addition and subtraction are componentwise. A scalar multiple

c v

scales the magnitude by

∣ c ∣

and reverses direction if

c < 0

. Geometrically, addition follows the **tip-to-tail** or parallelogram rule. The triangle inequality

∣ u + v ∣ \leq ∣ u ∣ + ∣ v ∣

holds, with equality only when

u

and

v

point in the same direction.

Key takeaways

Addition: $⟨ a_{1}, a_{2}, a_{3} ⟩ + ⟨ b_{1}, b_{2}, b_{3} ⟩ = ⟨ a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3} ⟩$ .
Scalar mult: $c ⟨ v_{1}, v_{2}, v_{3} ⟩ = ⟨ c v_{1}, c v_{2}, c v_{3} ⟩$ .
$∣ u + v ∣ \neq = ∣ u ∣ + ∣ v ∣$ in general.

Applications: forces, tensions, velocities

Resolve each given quantity into a vector with components along chosen axes, then add the vectors. The resultant magnitude and direction give the answer in physical terms. For an airplane heading at speed

∣ v ∣

in direction angle

θ

in 2D,

v = ⟨ ∣ v ∣ cos θ, ∣ v ∣ sin θ ⟩

Strategy tips

Always sketch the situation first — direction angles are slippery.
Add component-by-component, then take magnitude at the end.
Interpret the final answer in physical units.

Worked examples

Example 1

Given

P (1, 2, - 1)

and

Q (4, 0, 3)

, find

P Q

∣ P Q ∣

, and the unit vector in the direction of

P Q

1
Subtract initial from terminal.
$P Q = ⟨ 4 - 1, 0 - 2, 3 - (- 1)⟩ = ⟨ 3, - 2, 4 ⟩$
2
Magnitude.
$∣ P Q ∣ = 9 + 4 + 16 = 29$
3
Unit vector: divide each component by the magnitude.
$\hat{P Q} = ⟨ \frac{3}{29}, \frac{- 2}{29}, \frac{4}{29} ⟩$

Answer

P Q = ⟨ 3, - 2, 4 ⟩

∣ P Q ∣ = 29

, unit vector

= \frac{1}{29} ⟨ 3, - 2, 4 ⟩

Example 2

An airplane flies northwest at 550 mph and encounters a 40 mph crosswind from the south. Find the resultant velocity vector and ground speed.

1
Set up: east = $+ x$ , north = $+ y$ . Northwest means equal parts $- x$ and $+ y$ , so $θ = 135°$ .
$v_{plane} = ⟨ 550 cos 135°, 550 sin 135° ⟩ = ⟨ - 2752, 2752 ⟩$
2
Wind blows from south to north, so it has only a $+ y$ component.
$v_{wind} = ⟨ 0, 40 ⟩$
3
Add component-by-component.
$v = ⟨ - 2752, 2752 + 40 ⟩$
4
Compute magnitudes numerically.
$∣ v ∣ = (- 2752)^{2} + (2752 + 40)^{2} \approx 578.3 mph$

Answer

v \approx ⟨ - 388.9, 428.9 ⟩

mph; ground speed

\approx 578

mph.

Example 3

Find the unit vector that points from

A (2, - 1, 5)

toward

B (0, 3, 1)

1
Compute $A B$ .
$A B = ⟨ 0 - 2, 3 - (- 1), 1 - 5 ⟩ = ⟨ - 2, 4, - 4 ⟩$
2
Magnitude.
$∣ A B ∣ = 4 + 16 + 16 = 36 = 6$
3
Divide.
$\hat{u} = \frac{1}{6} ⟨ - 2, 4, - 4 ⟩ = ⟨ - \frac{1}{3}, \frac{2}{3}, - \frac{2}{3} ⟩$

Answer

\hat{u} = ⟨ - 1/3, 2/3, - 2/3 ⟩

Interactive visualizations

Loading visualization…

u_x2.00

u_y0.00

u_z0.00

v_x0.00

v_y2.00

v_z0.00

u \times v = ⟨ 0.00, 0.00, 4.00 ⟩

∣ u \times v ∣ = 4.00 = area of parallelogram

Three sliders per vector. Watch how the parallelogram changes as you move the tips. (Cross-product output preview—covered in detail later.)

Formulas in this topic

Vector magnitude

∣ v ∣ = v_{1}^{2} + v_{2}^{2} + v_{3}^{2}

Length of a 3D vector.

Vector addition

u + v = ⟨ u_{1} + v_{1}, u_{2} + v_{2}, u_{3} + v_{3} ⟩

Add componentwise.

Scalar multiple

c v = ⟨ c v_{1}, c v_{2}, c v_{3} ⟩

Scale each component by c.

Vector from points

P Q = ⟨ x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1} ⟩

Terminal minus initial.

Unit vector

\hat{v} = v /∣ v ∣

Direction with length 1.

2D vector from angle

v = ⟨ ∣ v ∣ cos θ, ∣ v ∣ sin θ ⟩

Build a vector from magnitude and direction.

Component form and i, j, k notation

From two points: terminal minus initial

Magnitude and unit vectors

Sums, differences, scalar multiples

Applications: forces, tensions, velocities

Worked examples

Interactive visualizations

Formulas in this topic

Related topics