Vectors in the plane describe quantities that have both size and direction, such as displacement, velocity, and force. A two-dimensional vector can be drawn as an arrow on a coordinate plane, with its tail at one point and its head at another. When the tail is at the origin and the head is at (a,b), the vector is written as <a,b>.
This notation connects geometry with algebra, making vectors useful for solving problems in math and physics.
The components of a vector tell how far it moves horizontally and vertically. Its magnitude comes from the Pythagorean theorem, and its direction can be described by an angle from the positive x-axis. Vectors can be added by adding components, and scaling a vector changes its length while keeping or reversing its direction.
The dot product connects two vectors to the angle between them and helps decide whether vectors point in similar, opposite, or perpendicular directions.
Key Facts
- A vector from the origin to (a,b) is written v = <a,b>.
- The magnitude of v = <a,b> is |v| = sqrt(a^2 + b^2).
- The direction angle θ of v = <a,b> satisfies tan θ = b/a, with quadrant checked from the signs of a and b.
- Vector addition is component-wise: <a,b> + <c,d> = <a + c, b + d>.
- Scalar multiplication is k<a,b> = <ka,kb>.
- The dot product is <a,b> · <c,d> = ac + bd = |v||w|cos θ.
Vocabulary
- Vector
- A quantity with both magnitude and direction, often represented by an arrow.
- Component
- One of the horizontal or vertical parts of a vector, such as a and b in <a,b>.
- Magnitude
- The length or size of a vector, found using the distance formula.
- Scalar
- A number that multiplies a vector to change its magnitude and possibly its direction.
- Dot product
- A multiplication operation on two vectors that produces a scalar related to the angle between them.
Common Mistakes to Avoid
- Adding magnitudes instead of components is wrong because vector addition depends on direction as well as length. Add x-components together and y-components together.
- Using tan θ = b/a without checking the quadrant is wrong because arctangent alone may give an angle in the wrong direction. Use the signs of a and b to place the angle correctly.
- Forgetting that a negative scalar reverses direction is wrong because multiplying by a negative number points the vector opposite the original direction. The magnitude is multiplied by the absolute value of the scalar.
- Confusing the dot product with vector addition is wrong because the dot product produces a single number, not a vector. Compute <a,b> · <c,d> as ac + bd.
Practice Questions
- 1 A vector v starts at the origin and ends at (6,8). Find v in component form and find |v|.
- 2 Let u = <3,-2> and w = <-5,4>. Find u + w, 2u, and u · w.
- 3 Two nonzero vectors have dot product 0. Explain what this tells you about the angle between them and how this would appear on a coordinate-plane diagram.