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
Back to Plane Vectors

Vector Operations Calculator

Calculate vector operations including magnitude, addition, subtraction, scalar multiplication, dot product, and check collinearity and perpendicularity relationships.

Vector Calculator
Choose calculation type and enter vector coordinates
Current Operation:
|x, y|
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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