MATH1061 Chap.6 Vectors: →, ·, ⟂
Vectors: →, ·, ⟂
Vectors encode magnitude and direction independently of a particular drawing. This chapter links component arithmetic to geometry through length, unit vectors, dot products, angles and projections. A projection is developed as the part of one vector parallel to another, while the residual is perpendicular. That decomposition supplies a built-in check and clarifies why the dot product determines both angles and orthogonality.
Coordinate formulas are always paired with the geometric object they compute. Vector equations distinguish a point from a direction and a displacement from an absolute position. Normalising removes scale but preserves direction, provided the original vector is nonzero. The sign of a dot product predicts whether an angle is acute, right or obtuse before inverse cosine is used.
Projection then turns that same product into a parallel component, leaving a residual whose zero dot product supplies an exact perpendicularity check. These relationships let a small sketch and a short calculation verify one another.
What this chapter covers
- 01
Represent vectors with components
- 02
Compute magnitude and unit direction
- 03
Use dot products to test angles
- 04
Recognise orthogonality from a zero dot product
- 05
Project one vector onto another
- 06
Decompose a vector into parallel and perpendicular parts
Worked example · free
Resolve a vector into two components
- 1Compute v dot u = 10 and u dot u = 5.
- 1The projection coefficient is 10/5 = 2.
- 1Therefore the projection is 2u = (4, 2).
- 1The residual is v - projection = (-1, 2).
- 1Its dot product with u is -2 + 2 = 0, confirming perpendicularity.
Key terms
- Unit vector
- A vector of length one that records direction without scale.
- Dot product
- A scalar product relating vector components, lengths and angle.
- Orthogonal vectors
- Vectors whose dot product is zero.
- Residual vector
- The component left after a projection is removed.
- Direction vector
- A nonzero vector specifying the orientation of a line or movement.
- Scalar component
- The signed numerical amount of one vector lying along a direction.
- Vector magnitude
- The nonnegative length computed from the square root of summed squared components.
- Normal vector
- A vector perpendicular to every direction lying within a plane.
Vectors: →, ·, ⟂ FAQ
Why does projection use a ratio of dot products?
The dot product with the target direction measures how much of the original vector points along it. Dividing by the target vector's squared length removes the scale already present in that direction. Multiplying by the target vector then reconstructs the parallel component with the correct magnitude and orientation.
What happens if the target vector is not a unit vector?
The general projection formula already handles that case by dividing by the target vector dotted with itself. If the target is a unit vector, that denominator equals one and disappears. Omitting the denominator for a non-unit target scales the projection incorrectly, even if its direction still looks plausible in a sketch.
How can I check a vector decomposition?
Use two checks. First, add the proposed parallel and perpendicular components and confirm that they reproduce the original vector. Second, take the dot product of the residual with the target direction and confirm that it is zero. A sketch can test orientation, but these algebraic checks establish the decomposition exactly.
How do I know whether a vector angle is obtuse?
Inspect the dot product before applying inverse cosine. A negative dot product means the cosine is negative, so the smaller angle between the nonzero vectors is obtuse. A positive value gives an acute angle and zero gives a right angle. This sign prediction also detects calculator mode or component errors in the final angle.
Exam move
Keep a small geometric sketch beside every component calculation. Label the original vector, the target direction, the projection and the residual. Begin with quick component drills for addition, displacement and length, then move to questions where the target direction is not a unit vector. Estimate whether the projection should point with or against that direction before applying the formula.
Write the scalar projection coefficient separately from the projected vector so their roles do not blur. After calculation, perform both reconstruction and orthogonality checks. For angle questions, predict acute, right or obtuse from the sign of the dot product before using inverse cosine, and verify that the cosine ratio remains between negative one and one.
For line or plane geometry, distinguish the fixed point from the direction or normal vector before substituting coordinates. When an answer is wrong, redraw the arrows and identify whether the error came from orientation, scaling, a component product or the final interpretation.
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