Vector Algebra · Lesson 2 of 9

Adding and Scaling Vectors

“Build vector sums and differences geometrically, and see how scalar multiplication changes an arrow.”

Learning Objectives

• Construct a vector sum using the triangle and parallelogram laws. • Explain vector subtraction and a zero resultant around a closed path. • Justify commutative and associative addition using diagrams. • Interpret positive, negative and zero scalar multiples and construct a unit vector.

Successive displacements

If you walk from $A$ to $B$ and then from $B$ to $C$ , your final displacement is from $A$ to $C$ . The total distance walked may be longer than this direct displacement. Vector addition describes the change in position, including direction.

Triangle lawLaTeX

A B + B C = A C

Place the second vector with its tail at the head of the first. The resultant runs from the first tail to the final head.

Head-to-tail vector addition— The blue and teal arrows form the route. The orange arrow is the resultant.

Free vectors may be translated before addition, provided their lengths and directions remain unchanged. Rotating an arrow to make it fit would change the vector. This is why head-to-tail construction is possible even when the original arrows were drawn at unrelated places.

Example — A short walk

Problem
A person walks 3 km east and then 4 km north. Find the resultant displacement.

1.Draw the eastward and northward arrows head to tail. They form the perpendicular sides of a right triangle.
2.The magnitude of the direct displacement is $3^{2} + 4^{2} = 5$ km.
3.The direction is north of east, with $tan θ = 4/3$ . The distance walked is 7 km, but the displacement magnitude is 5 km.

Closed paths and subtraction

A journey that returns to its starting point has zero displacement. This observation gives useful identities without requiring any lengths or angles. Subtraction then follows by replacing a vector with its negative.

A closed triangular pathLaTeX

A B + B C + C A = 0

The starting and finishing points coincide. The identity also holds when the three points are collinear.

To form $a - b$ , add $- b$ to $a$ . When $a$ and $b$ have a common tail, their difference points from the head of $b$ to the head of $a$ . This direction is worth checking before calculating.

Subtraction of coinitial vectors— From B, the head of b, to A, the head of a, the arrow is a − b.

Example — Identifying a difference

Problem
If $a = O A$ and $b = O B$ , express $B A$ .

1.Travel from $B$ to $O$ and then from $O$ to $A$ : $B A = B O + O A$ .
2.Since $B O = - b$ , the answer is $a - b$ .
3.Reversing the requested direction would give $A B = b - a$ .

Two vectors acting together

The triangle law describes successive displacements, but vectors can also act simultaneously. A boat moving across a river has velocity relative to the water while the water itself flows downstream. Their sum is the boat’s velocity relative to the bank.

Definition

Parallelogram law

Draw two coinitial vectors as adjacent sides of a parallelogram. The diagonal beginning at their common tail represents their sum in both magnitude and direction.

The parallelogram law— Either two-edge route gives the same diagonal resultant.

The opposite sides are equal free vectors. Therefore the diagonal can be reached by adding $a$ and then $b$ , or by adding $b$ and then $a$ . The parallelogram law and triangle law are two views of the same addition.

Example — Crossing a flowing river

Problem
A boat moves north through still water at 4 m/s, while the river flows east at 3 m/s. Describe its velocity relative to the bank.

1.Draw the northward boat velocity and eastward water velocity as adjacent sides of a parallelogram.
2.The diagonal points northeast. Its magnitude is $4^{2} + 3^{2} = 5$ m/s.
3.The direction makes an angle $θ$ east of north with $tan θ = 3/4$ . It is not travelling straight north relative to the bank.

The algebra of vector addition

A vector sum has a definite result independent of the order in which we carry out its construction. The diagrams explain the algebraic rules rather than merely providing pictures to memorise. These rules let us simplify a longer chain of displacements confidently.

Commutative and associative additionLaTeX

a + b = b + a, (a + b) + c = a + (b + c)

Order and grouping do not change a sum. The second statement follows because both groupings run from the same starting point to the same final point in a head-to-tail chain.

For associativity, draw three arrows head to tail. Replacing the first two by their resultant leaves the same start and finish as replacing the last two first. The zero vector is the additive identity, while $- a$ is the additive inverse of $a$ .

Identity and inverseLaTeX

a + 0 = a, a + (- a) = 0

An unchanged position adds no displacement; an equal reverse displacement cancels the original one.

Multiplying an arrow by a scalar

Scalar multiplication changes the scale of a vector. A positive factor preserves its direction, a negative factor reverses it, and a factor of zero gives the zero vector. The absolute value of the scalar determines the new length.

Magnitude after scalingLaTeX

∣ λ a ∣ = ∣ λ ∣ ∣ a ∣

λ

is a real scalar. The magnitude is nonnegative even when

λ < 0

Scaling and reversal— The reference arrow a points right. Negative multiples point left.

Dividing a nonzero vector by its own magnitude shrinks or stretches it to length one without changing direction. We use a hat to denote the resulting unit vector. This construction cannot be applied to the zero vector.

Normalising a vectorLaTeX

\overset{a}{^} = \frac{a}{∣ a ∣} (a \neq = 0)

Its magnitude is

∣ a ∣/∣ a ∣ = 1

. A vector of magnitude

M > 0

in the same direction is

M \overset{a}{^}

Example — Scaling a displacement

Problem
A displacement $a$ has magnitude 6 m. Describe $- \frac{3}{2} a$ and find a unit vector along $a$ .

1.The new magnitude is $∣ - 3/2∣ \times 6 = 9$ m.
2.The negative factor makes its direction opposite to the original displacement.
3.The unit vector is $\overset{a}{^} = a /6$ when components are expressed in the chosen metre scale. Scaling this unit direction by a required magnitude constructs a new vector along the same line.

Common mistake

In general, $∣ a + b ∣$ is not $∣ a ∣ + ∣ b ∣$ . Add the vectors with their directions first. Also, $a - b$ and $b - a$ point in opposite directions unless both are zero.

Quick check

Which arrow is $A B + B C$ ?

Quick check

What is $A B + B C + C A$ ?

Quick check

For coinitial vectors, the sum is represented by

Quick check

If $∣ a ∣ = 5$ , then $∣ - 2 a ∣$ is

Quick check

What does associativity allow?

Practice Problems

Draw $a + b$ and $a - b$ for two nonparallel coinitial arrows. Label their directions clearly.
Use a head-to-tail construction to show why adding three vectors is associative.
A traveller goes 6 km east and 8 km north. Find the magnitude and describe the direction of the resultant displacement.
Simplify $A B + B C - A C$ and explain geometrically.
If $∣ p ∣ = 8$ , find the magnitudes and directions relative to $p$ of $3 p$ , $- p /4$ and $0$ .
Explain why the velocity of a boat relative to the bank differs from its velocity relative to the water in a flowing river.

Key Takeaways

Head-to-tail addition and the parallelogram diagonal describe the same resultant. A closed path has zero resultant; subtraction means addition of the negative vector. Vector addition is commutative and associative. Scalar multiplication scales length by the absolute value of the factor and may reverse direction. A nonzero vector divided by its magnitude is a unit vector.

Previous · Lesson 1

Understanding Vectors in Space

Next · Lesson 3

Vector Components and Direction

Successive displacements

Closed paths and subtraction

Two vectors acting together

The algebra of vector addition

Multiplying an arrow by a scalar

Quiz

Practice Problems

Key Takeaways