A vector describes both size and direction, making it useful for motion, forces, displacement, and many other quantities in physics and mathematics. In two dimensions, a vector like v = ⟨3, 4⟩ can be drawn as an arrow from the origin to the point (3, 4). Its magnitude is the length of that arrow, found using the Pythagorean theorem.
For v = ⟨3, 4⟩, the magnitude is |v| = 5, forming the classic 3-4-5 right triangle.
A unit vector keeps only the direction of a vector while changing its length to exactly 1. To normalize a vector, divide each component by the vector's magnitude, so v/|v| = ⟨3/5, 4/5⟩ for v = ⟨3, 4⟩. Standard basis vectors i = ⟨1, 0⟩ and j = ⟨0, 1⟩ let us write vectors as component sums such as v = 3i + 4j.
Direction angles connect vectors to trigonometry because the x and y components can be found from magnitude and angle using cosine and sine.
Key Facts
- For v = ⟨a, b⟩, the magnitude is |v| = sqrt(a^2 + b^2).
- For v = ⟨3, 4⟩, |v| = sqrt(3^2 + 4^2) = 5.
- A unit vector in the direction of v is u = v/|v|, as long as v is not the zero vector.
- Normalizing ⟨3, 4⟩ gives u = ⟨3/5, 4/5⟩.
- The standard basis vectors are i = ⟨1, 0⟩ and j = ⟨0, 1⟩, so ⟨a, b⟩ = ai + bj.
- For a vector with magnitude r and direction angle θ from the positive x-axis, v = ⟨r cos θ, r sin θ⟩.
Vocabulary
- Vector
- A vector is a quantity with both magnitude and direction, often written using components such as ⟨a, b⟩.
- Magnitude
- Magnitude is the length or size of a vector, written as |v|.
- Unit vector
- A unit vector is a vector with magnitude 1 that indicates direction only.
- Component
- A component is one coordinate part of a vector, such as the x-value a or y-value b in ⟨a, b⟩.
- Standard basis vector
- A standard basis vector is a unit vector along a coordinate axis, such as i = ⟨1, 0⟩ or j = ⟨0, 1⟩.
Common Mistakes to Avoid
- Adding components to find magnitude is wrong because vector length uses the Pythagorean theorem, not a + b. For example, |⟨3, 4⟩| is 5, not 7.
- Forgetting to divide every component when normalizing is wrong because the whole vector must be scaled by the same factor. The unit vector for ⟨3, 4⟩ is ⟨3/5, 4/5⟩, not ⟨3/5, 4⟩.
- Normalizing the zero vector is wrong because |⟨0, 0⟩| = 0 and division by zero is undefined. The zero vector has no unique direction.
- Using tan θ = y/x without checking the quadrant can give the wrong direction angle. The signs of both components must be used to place the angle in the correct quadrant.
Practice Questions
- 1 Find the magnitude of a = ⟨-6, 8⟩ and write a unit vector in the same direction.
- 2 A vector has magnitude 10 and direction angle 30° from the positive x-axis. Find its x and y components in exact form.
- 3 Two vectors have the same unit vector but different magnitudes. Explain what is the same about the vectors and what is different.