Vectors and Their Applications
FoundationVectors carry both magnitude and direction: from magnitude, unit vectors and coordinate form, through the dot product for angles and orthogonality and the cross product for areas, to applications in dividing segments, testing parallelism and projections across two and three dimensions.
- Vectors have magnitude and direction: |a|, the unit vector a/|a|, and the directionless zero vector; notation differs between print and writing.
- Add and subtract by the triangle rule; parallelism is a = kb; coordinates turn geometry into algebra with the 3D Pythagorean magnitude.
- The dot product is a number: a·b = |a||b|cos θ = the component sum; zero dot product means orthogonal — used for angles and perpendicularity proofs.
文體小知識：數學概念說明文——由向量與純量的概念、運算與坐標表示走到點積與叉積，再以夾角、投影、線段分割與三角形面積示範向量的幾何應用。
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1A quantity with size only is a scalar; one with both size and direction is a vector — displacement, force and velocity are vectors. Vectors are printed in bold, written with an arrow, or named by two points running from A towards B; their length is the magnitude. A unit vector has magnitude one and the zero vector has magnitude zero with no direction. Dividing any non-zero vector by its own magnitude gives the unit vector of the same direction — the basic way to express direction.
2Addition follows the triangle rule: attach the start of b to the end of a and draw from the start of a to the end of b; subtraction must not reverse the direction. Scalar multiplication scales the magnitude and reverses direction for negative scalars; two non-zero vectors are parallel if and only if one equals the other times a scalar. In rectangular coordinates a vector is written by components; the magnitude generalises Pythagoras to three dimensions, and direction angles come from component-to-magnitude ratios.
3The dot product presses two vectors into a number: |a||b|cos θ, or the sum of the products of corresponding components. Its greatest uses are finding angles and testing orthogonality: two non-zero vectors are perpendicular if and only if their dot product is zero. For example, the work done by a force is the dot product of force and displacement — which is why a force perpendicular to the motion does no work; to prove two lines perpendicular, prove the dot product of their direction vectors is zero.
4Three-dimensional space also has the cross product: a×b is a vector perpendicular to the plane of a and b, directed by the right-hand rule, with magnitude |a||b|sin θ — exactly the area of the parallelogram spanned by the two vectors. Cross products are computed by expanding a three-by-three determinant, are anti-commutative, and any vector crossed with itself is zero; the cyclic order of i, j, k is the key to the computation.
5Applications cluster in geometry: angles from the dot product formula; projections resolving one vector along another; division of a line segment by the section formula, locating a point in a given ratio; and the area of a triangle as half the magnitude of the cross product of two sides. Tests of parallelism and orthogonality run through every application — the shared starting point of proofs and computations, and the two keys to M2 vector questions.
1Summary 重點摘要
Vectors carry both magnitude and direction: from magnitude, unit vectors and coordinate form, through the dot product for angles and orthogonality and the cross product for areas, to applications in dividing segments, testing parallelism and projections across two and three dimensions.
2Vocabulary cards Vocabulary
A quantity with both magnitude and direction, such as displacement, force or velocity.
A quantity with size only and no direction, such as temperature, mass or time.
The length of a vector, written |a|, the square root of the sum of squared components.
A vector of magnitude one; dividing a non-zero vector by its magnitude gives the unit vector of the same direction.
The vector from the origin to a point P, with components equal to the coordinates of P.
a·b = |a||b|cos θ = the sum of products of corresponding components; the result is a number.
In three dimensions, a×b is a vector perpendicular to both, with magnitude |a||b|sin θ.
The signed length (scalar projection) or the component vector (vector projection) of one vector along another.
Two non-zero vectors at right angles, exactly when their dot product is zero.
The formula locating a point dividing a segment in a given ratio; the midpoint formula is a special case.
3Step by step Steps
- Step 1: Vectors versus scalars, notation and magnitudeA quantity with size only is a scalar — temperature, mass, time; one with both size and direction is a vector — displacement, force, velocity. Notation runs in two systems: bold letters a in print, an arrow over the letter in writing, or two points naming the vector from A towards B; the magnitude is written |a|. Two special vectors: the unit vector of magnitude one, and the zero vector of magnitude zero with no direction. For example, dividing a non-zero vector by its own magnitude gives the unit vector of the same direction — exactly how a force is split into direction and size.
- Step 2: Operations on vectorsAddition uses the triangle rule: attach the start of b to the end of a and draw from the start of a to the end of b; for a − b, draw from the end of b to the end of a — the direction must not be reversed. Scalar multiplication scales the magnitude, reversing direction for negative scalars; two non-zero vectors are parallel if and only if a = kb. The algebra mirrors numbers: commutative, associative, distributive. For example, walking three kilometres north then four east gives a resultant of magnitude five — the triangle rule with Pythagoras; opposite forces cancel.
- Step 3: Rectangular coordinate formIn coordinates, the vector from the origin to a point P is the position vector of P, with components equal to P's coordinates. A two-dimensional vector a = xi + yj has magnitude √(x²+y²); adding zk gives the three-dimensional magnitude √(x²+y²+z²). Direction angles come from component-to-magnitude ratios: sin θ = y/√(x²+y²) and cos θ = x/√(x²+y²). For example, 3i + 4j has magnitude five, with cosine three fifths and sine four fifths of the angle from the x-axis. Coordinates turn geometry into algebra; the curriculum excludes direction cosines, so no need to go deeper.
- Step 4: The scalar product (dot product)The dot product presses two vectors into a number: a·b = |a||b|cos θ, or the sum of the products of corresponding components. It is commutative, distributive over addition, and a·a = |a|². The crucial fact: two non-zero vectors are perpendicular if and only if their dot product is zero, since the cosine of ninety degrees vanishes. For example, work done is the dot product of force and displacement — a force perpendicular to the motion does no work; to prove two lines perpendicular, write their direction vectors and show the dot product is zero — the standard opening of proof questions.
- Step 5: The vector product (cross product)The three-dimensional cross product a×b is a vector: perpendicular to the plane of a and b, directed by the right-hand rule, with magnitude |a||b|sin θ — precisely the area of the parallelogram they span. Compute it by a three-by-three determinant: i, j, k in the first row, the components of the two vectors in the second and third rows, then expand. Properties include the anti-commutative law b×a = −(a×b) and a×a = 0. For example, i×j = k, j×k = i, k×i = j — a broken cyclic order ruins the signs, and the minus before the j component is the easiest thing to drop: the two classic cross-product slips.
- Step 6: Angles and projectionsSolve the dot product definition for the angle: cos θ = a·b ÷ (|a||b|) — compute the dot product and both magnitudes, then apply the inverse cosine. Projections come in two kinds: the scalar projection of a along b is the signed length a·b ÷ |b|; the vector projection multiplies further by the unit vector of b, giving the actual component vector. For example, a = 2i + j + 2k and b = i + 2j + 2k: the dot product is 8, both magnitudes are 3, cos θ = 89 and the angle is about 27.3 degrees; the scalar projection of a onto b is 83. A positive dot product signals an acute angle — a quick sanity check.
- Step 7: Geometric applications: division, parallelism and areaDivide a segment with the section formula: if AP:PB = m:n, the position vector of P equals (n × OA + m × OB) ÷ (m+n); the midpoint formula is the case m = n. Parallelism is tested by a = kb and orthogonality by a zero dot product — the two great tools of geometric proof. The area of a triangle is half the magnitude of the cross product of its two side vectors. For example, the triangle A(0, 0), B(4, 0), C(0, 3) has side vectors AB = 4i and AC = 3j; the cross product's magnitude is twelve and the area six — agreeing with base times height over two, confirming the formula.
5Worked example Example
Example: given a = 2i + j + 2k and b = i + 2j + 2k, find the angle between a and b, and the projection of a onto b.
- Magnitudes: |a| = √(2² + 1² + 2²) = √9 = 3; |b| = √(1² + 2² + 2²) = √9 = 3
- Dot product: a·b = 2×1 + 1×2 + 2×2 = 8
- Angle: cos θ = a·b ÷ (|a||b|) = 89, so θ = cos⁻¹(89) ≈ 27.3°
- Projections: scalar projection = a·b ÷ |b| = 83; vector projection = (89)b = (89)(i + 2j + 2k)
- Answer: The angle is about 27.3°, the scalar projection of a onto b is 83, and the vector projection is (89)(i + 2j + 2k). Marking points: correct magnitudes and dot product, the angle formula, and the projection definitions with the unit vector; a positive dot product confirms an acute angle — a quick check.
6Common mistakes Mistakes
7Quick check Quick check
Q1How do you find a unit vector in the direction of a non-zero vector a?
Q2Find the magnitude of the vector (1, 2, 2).
Q3Are a = (2, −1, 3) and b = (1, 2, 0) orthogonal?
Q4For a = (1, 2) and b = (3, 4), write down cos θ for the angle between them.
Q5How is the area of a triangle found with the cross product?
8Recap Recap
- Vectors have magnitude and direction: |a|, the unit vector a/|a|, and the directionless zero vector; notation differs between print and writing.
- Add and subtract by the triangle rule; parallelism is a = kb; coordinates turn geometry into algebra with the 3D Pythagorean magnitude.
- The dot product is a number: a·b = |a||b|cos θ = the component sum; zero dot product means orthogonal — used for angles and perpendicularity proofs.
- The cross product is a vector: magnitude |a||b|sin θ is the parallelogram area, direction by the right-hand rule, determinant signs alternating.
- Four applications: angles, projections, the section formula for dividing segments, and triangle area = |a×b| ÷ 2.