MathIsimple

Plane Vectors

Master 2D vectors: concepts, operations, coordinate representations, and the powerful dot product.

Geometry
Intermediate
~60 min
Vector Calculator
Calculate magnitude, add, subtract, scale vectors, and check collinearity/perpendicularity

Enter vectors to see operations

1. What is a Vector?

A vector is a quantity with both magnitude (size) and direction.

Notation

  • a⃗\vec{a} or a (bold) — vector a
  • ∣a⃗∣|\vec{a}| or ∥a⃗∥\|\vec{a}\| — magnitude of a
  • AB⃗\vec{AB} — vector from A to B

Special Vectors

  • Zero vector 0⃗\vec{0}: magnitude 0, no direction
  • Unit vector: magnitude = 1
  • Standard basis: i^=(1,0)\hat{i} = (1,0), j^=(0,1)\hat{j} = (0,1)
2. Vector Operations
3. Coordinate Representation
4. Dot Product
5. Applications

Practice Quiz

Practice Quiz
20
Questions
0
Correct
0%
Accuracy
1
A vector has:
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2
If a⃗=(3,4)\vec{a} = (3, 4), what is ∣a⃗∣|\vec{a}|?
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3
If a⃗=(2,3)\vec{a} = (2, 3) and b⃗=(1,−1)\vec{b} = (1, -1), what is a⃗+b⃗\vec{a} + \vec{b}?
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4
If a⃗=(4,6)\vec{a} = (4, 6), what is 2a⃗2\vec{a}?
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5
The dot product a⃗⋅b⃗\vec{a} \cdot \vec{b} is:
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6
If a⃗=(3,4)\vec{a} = (3, 4) and b⃗=(2,1)\vec{b} = (2, 1), what is a⃗⋅b⃗\vec{a} \cdot \vec{b}?
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7
Two vectors are perpendicular if their dot product is:
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8
The zero vector 0⃗\vec{0} has:
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9
If a⃗=(1,0)\vec{a} = (1, 0), what type of vector is this?
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10
The vector from point A(1, 2) to B(4, 6) is:
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11
a⃗⋅a⃗\vec{a} \cdot \vec{a} equals:
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12
If a⃗=(6,8)\vec{a} = (6, 8), the unit vector in the direction of a⃗\vec{a} is:
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13
If a⃗⋅b⃗=∣a⃗∣∣b⃗∣\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|, the angle between them is:
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14
If a⃗⋅b⃗=−∣a⃗∣∣b⃗∣\vec{a} \cdot \vec{b} = -|\vec{a}||\vec{b}|, the angle between them is:
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15
The formula for the dot product is:
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16
If a⃗=(1,2)\vec{a} = (1, 2) and b⃗=(−2,1)\vec{b} = (-2, 1), then a⃗⋅b⃗\vec{a} \cdot \vec{b} is:
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17
The projection of a⃗\vec{a} onto b⃗\vec{b} has direction:
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18
If a⃗=(3,0)\vec{a} = (3, 0) and b⃗=(0,4)\vec{b} = (0, 4), then ∣a⃗+b⃗∣|\vec{a} + \vec{b}| is:
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19
Scalar multiplication changes:
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20
The angle θ\theta between two vectors can be found using:
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