Motion in a Plane · Lesson 2 of 13
Vector Addition, Subtraction and Scaling
“Build vector sums graphically and use zero vectors, differences and magnitude bounds correctly.”
• Add vectors by the triangle and parallelogram methods. • Subtract a vector by adding its opposite. • Explain scalar multiplication, zero vectors and vector algebra laws. • Derive magnitude inequalities and their equality cases.
Multiplying an arrow by a real number
Multiplying a vector by 2 doubles its magnitude while keeping its direction. Multiplication by a positive fraction shortens it. Multiplication by a negative number reverses direction as well as changing magnitude. For any real number λ, the magnitude of λA is |λ| times the magnitude of A. The absolute value is essential: a vector’s magnitude cannot become negative.
A scalar multiplier can also carry units. A constant velocity vector multiplied by elapsed time gives a displacement vector. Its physical dimensions are the product of the scalar’s dimensions and the vector’s dimensions. Adding physical quantities is more restrictive: quantities must have compatible units and meanings. You cannot add a mass to a temperature, a velocity to an acceleration, or a scalar speed directly to a velocity vector.
The head-to-tail rule
To add A and B, draw A, then move a parallel copy of B until its tail meets A’s head. Draw the resultant from A’s original tail to B’s final head. This is the triangle law. The resultant represents one overall effect of the two successive displacements. Measuring a scale drawing gives an estimate; components will later provide a more accurate calculation.
The parallelogram method begins with A and B at a common tail. Through each head draw a line parallel to the other vector. The diagonal from the common tail to the opposite corner is A + B. It gives the same answer as head-to-tail addition because opposite sides of a parallelogram are equal parallel vectors. The other diagonal connects the two heads and represents a difference, with the direction depending on which head you start from.
Order and grouping do not change a sum
These laws allow a long sequence of displacements to be added in any order and grouped conveniently. They do not mean every physical journey follows the same route; only the final resultant displacement stays the same. A polygon of head-to-tail arrows closes exactly when their vector sum is zero.
Zero vectors and subtraction
Subtraction is not commutative: reversing the order reverses the answer. A zero resultant does not imply that every contributing vector is zero. Two equal opposite displacements cancel, and three nonzero vectors can form a closed triangle. Three noncoplanar vectors cannot sum to zero: if A + B + C = 0, then C = −(A + B) must lie in the plane of A and B, or their common line if they are collinear.
How large can a resultant be?
Walking directly from start to finish cannot require more distance than following two successive straight segments. Applied to A and B, this gives |A + B| ≤ A + B. Rearranging A = (A + B) − B and applying the same upper bound gives A ≤ |A + B| + B, hence |A + B| ≥ A − B. Interchanging A and B supplies the other sign and combines the two into a lower bound.
Problem
Two vectors have magnitudes 7 and 3 units. What magnitudes are possible for their sum?
- 1.The lower bound is |7 − 3| = 4 and the upper bound is 7 + 3 = 10.
- 2.Any resultant magnitude between 4 and 10 is possible by varying their angle. Values 2 or 12 are impossible.
- 3.If the sum is zero, magnitudes must be equal and directions opposite; these unequal vectors cannot cancel.
Reasoning from a closed set of vectors
Problem
Suppose A + B + C + D = 0. What follows without knowing the individual arrows?
- 1.The vectors need not individually vanish; a closed quadrilateral provides a nonzero example.
- 2.A + C = −(B + D), so the two grouped sums have equal magnitudes.
- 3.A = −(B + C + D), so |A| ≤ |B| + |C| + |D| by repeated triangle inequalities.
- 4.B + C = −(A + D), so B + C lies in the plane spanned by A and D, or on their line if those vectors are collinear.
Quiz
Two nonzero vectors have a zero sum. What must be true?
Key Takeaways
• Scale magnitudes with |λ| and reverse direction for negative λ. • Add arrows head to tail or use a parallelogram. • Subtract by adding the opposite arrow. • A closed polygon has zero vector sum. • A resultant magnitude lies between the difference and sum of the original magnitudes.