MAT188H1 Chap.1 Vectors and Scalar Multiples
Vectors and Scalar Multiples
Position and displacement answer different questions
A point records a location, while a vector records a displacement. The distinction matters even when both are written using coordinates. To travel from a point P to a point Q, subtract each coordinate of P from the corresponding coordinate of Q. The resulting vector tells you how to move from the starting point to the endpoint.
Reversing the journey changes every component's sign. Moving an arrow without changing its length or direction does not change the displacement it represents.
Standard position gives this displacement a convenient reference: place the tail at the origin. Its tip then determines a point whose coordinates agree with the vector's components.
This correspondence allows a set of vectors to describe a geometric object such as a line. It does not make an arbitrary point into a direction. When a problem supplies a point and an arrow, label their roles before combining their coordinates.
Components retain their order
In this course a vector normally means a column vector. A vector in Rn has n real components, arranged in a fixed order.
The first component cannot be exchanged with the second without potentially changing the vector. A row vector has a different written shape, even when the entries are the same. Dimension is therefore part of the information you should retain from the beginning of a calculation.
Vector addition pairs corresponding components. Both inputs must have the same number of components, and the output has that same number.
Scalar multiplication applies one real number to every component, including any zero components. These are separate operations: addition combines two vectors, while scalar multiplication combines a scalar and a vector.
Do not use ordinary multiplication between two vectors unless a particular operation has been defined.
Arithmetic and the arrow diagram agree
To draw a sum, place the tail of the second arrow at the tip of the first. The sum runs from the original starting point to the final endpoint. In the parallelogram picture, the same sum is the diagonal from the common tail.
A component calculation and a drawing express the same movement in different ways. If their endpoints disagree, inspect coordinate order and negative signs before changing the geometric interpretation.
Multiplication by a positive scalar preserves a nonzero vector's direction. A negative scalar reverses it, and its absolute value controls the change in length. Multiplication by zero gives the zero vector.
Subtraction can therefore be handled as addition of the opposite vector: reverse the vector being subtracted, then use the head to tail rule. This keeps the minus sign attached to the whole vector rather than to only its first entry.
One multiplier must satisfy every component
Parallel vectors are characterized by scalar multiplication.
To test a proposed relationship, find a candidate multiplier from one nonzero component and check the remaining components against that same multiplier. Agreement in a single coordinate proves too little.
If a component of the starting vector is zero, multiplication keeps it zero, so a nonzero target entry in that position immediately rules out the proposed relationship.
The definition permits one vector to be a scalar multiple of the other, so the zero vector needs careful treatment. It is a zero multiple of every vector, but it cannot serve as the nonzero direction required to describe a line.
Algebraic parallelism and the choice of a usable direction vector are related questions with different conditions. Write the relevant condition beside your conclusion instead of relying on a sketch of an arrow of zero length.
Use the vector algebra rules to simplify expressions without losing their type.
Distributing a scalar across a sum, gathering scalar multiples of the same vector, and changing the order of addition are legitimate. The final result should still have the original dimension. A useful finishing check is to calculate one component independently from the unsimplified expression and compare it with your final answer.
What this chapter covers
- 01
Points, displacement and standard position
- 02
Column vectors and componentwise operations
- 03
Scalar multiples and parallelism
Worked example · free
A displacement and its opposite journey
- 2Subtract starting coordinates from endpoint coordinates: Q - P = (-4 - 2, 5 - (-3), 7 - 1) = (-6, 8, 6). Each entry describes one part of the journey.
- 2The first component proposes a multiplier of 9/(-6) = -3/2. Test it rather than choosing separate multipliers for the other coordinates: (-3/2)8 = -12 and (-3/2)6 = -9.
- 2All components satisfy w = (-3/2)(Q - P), so the vectors are parallel. The negative multiplier reverses direction, while its absolute value increases length by a factor of 3/2.
Key terms
- Column Vector
- A matrix with one column whose ordered entries are its components. In Rⁿ those entries are real numbers, and the number of entries identifies the dimension in which the vector is being used.
- Position Vector
- The vector from the origin to a specified point. Its components match the point coordinates, providing the correspondence used to describe geometric sets through vectors in standard position.
- Scalar Multiple
- A vector obtained by multiplying every component of another vector by the same real number. The scalar must act uniformly across the whole vector, including entries equal to zero.
- Parallel Vectors
- Vectors for which one is a scalar multiple of the other. A valid component test uses one common multiplier; separate ratios cannot be chosen independently to force a relationship.
Vectors and Scalar Multiples FAQ
Can a vector start somewhere other than the origin?
Yes. An arrow can represent the same displacement after translation to a different starting location. Its endpoint then changes, but its components remain fixed. Standard position is a convenient convention that makes the endpoint coordinates equal the displacement components; it is not a restriction on where an actual journey begins.
Why does a zero component make ratio tests awkward?
Dividing by that component is undefined. Instead, inspect the component equation directly: multiplying zero by any scalar still gives zero. Use a nonzero entry to determine a possible scalar, then substitute that value throughout the vector. This method also exposes a target component that cannot possibly match.
Does reversing an arrow change whether it is parallel?
Reversal multiplies a vector by negative one, which preserves the scalar multiple relationship. It changes orientation along the same direction line. Distinguish the statement that vectors are parallel from the stronger statement that nonzero arrows point in the same direction; the latter requires a positive multiplier.
Exam move
Alternate coordinate calculations with small arrow sketches. For each subtraction, label the start and endpoint before computing. For each parallelism question, record one candidate scalar and test every component. Include examples with a zero entry so that the method remains reliable when taking component ratios is impossible.
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