Add vectors end-to-end, by the parallelogram rule and component-wise, and subtract by adding the additive inverse

Learning goal: Add vectors end-to-end, component-wise and by the parallelogram rule; understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes; given two vectors in magnitude and direction form, determine the magnitude and direction of their sum; and understand vector subtraction v − w as v + (−w), where −w is the additive inverse of w, representing subtraction graphically by connecting the tips in the appropriate order and performing it component-wise.

Scope note. N.VM.4, with its parts a, b and c, is marked (+) in Ohio's Learning Standards for mathematics, which Ohio uses to indicate a standard "for students who are planning on taking advanced courses". It is taught here in full, as extension content for students on that path.

Three pictures of the same sum. Let u = ⟨4, 1⟩ and w = ⟨1, 3⟩. The sum u + w can be found three ways, and they must agree.

End-to-end (tip-to-tail). Draw u from the origin: it ends at (4, 1). Now draw w starting at the tip of u: from (4, 1) move 1 right and 3 up to (5, 4). The sum is the arrow from the tail of the first vector to the tip of the last, from (0, 0) to (5, 4). This picture matches the meaning of a sum: do one displacement, then the other, and ask where you ended up.

Parallelogram rule. Draw both u and w from the same initial point, the origin. Complete the parallelogram whose two sides are these arrows; its fourth vertex is at (5, 4). The sum is the diagonal from the common initial point to the opposite vertex. The parallelogram rule and the end-to-end rule are the same construction seen twice: the side of the parallelogram opposite w is a copy of w placed at the tip of u.

Component-wise. Add horizontal components together and vertical components together: ⟨4 + 1, 1 + 3⟩ = ⟨5, 4⟩. This is the version you compute with; the pictures are the version you check with.

A fully worked model, with magnitudes. |u| = √(4² + 1²) = √17 ≈ 4.123. |w| = √(1² + 3²) = √10 ≈ 3.162. Their sum of magnitudes is about 7.285. The sum vector ⟨5, 4⟩ has magnitude √(25 + 16) = √41 ≈ 6.403. So |u + w| ≈ 6.40 while |u| + |w| ≈ 7.29. The magnitude of a sum of two vectors is typically not the sum of the magnitudes. Geometrically the three vectors form a triangle with sides √17, √10 and √41, and one side of a triangle is shorter than the other two put together. The general fact is |u + w| ≤ |u| + |w|, with equality only when u and w point in the same direction (or one of them is the zero vector). For instance ⟨2, 1⟩ and ⟨4, 2⟩ point the same way; their magnitudes are √5 and √20 = 2√5, and the sum ⟨6, 3⟩ has magnitude √45 = 3√5, exactly the sum. At the other extreme, ⟨3, 4⟩ + ⟨−3, −4⟩ = ⟨0, 0⟩, magnitude 0, though each part has magnitude 5.

Sums given in magnitude and direction form. Suppose two vectors are given by magnitude and direction: a has magnitude 10 at 0° (along the positive x-axis) and b has magnitude 10 at 60°. To determine the magnitude and direction of their sum, convert to components, add, and convert back. a = ⟨10 cos 0°, 10 sin 0°⟩ = ⟨10, 0⟩. b = ⟨10 cos 60°, 10 sin 60°⟩ = ⟨5, 5√3⟩ ≈ ⟨5, 8.660⟩. Sum: ⟨15, 5√3⟩. Magnitude: √(15² + (5√3)²) = √(225 + 75) = √300 = 10√3 ≈ 17.32. Direction: arctan(5√3/15) = arctan(√3/3) = 30° exactly (with the rounded 8.660, arctan(8.660/15) ≈ 30.00°). The direction is exactly halfway between 0° and 60° because the two magnitudes are equal; and 17.32 is less than 10 + 10 = 20, as expected for vectors 60° apart. Check by converting back: ⟨17.32 cos 30°, 17.32 sin 30°⟩ = ⟨15.0, 8.66⟩.

The additive inverse. For any vector w, the vector −w is its additive inverse: it has the same magnitude as w, it points in the opposite direction whenever w is not the zero vector (−0 = 0, which has no direction), and w + (−w) = 0, the zero vector. In components, −⟨1, 3⟩ = ⟨−1, −3⟩. Reversing an arrow does not change its length: |−w| = |w|.

Subtraction. Vector subtraction is defined by v − w = v + (−w): to subtract w, add its additive inverse. Component-wise this means subtracting each component: ⟨v_x − w_x, v_y − w_y⟩. With v = ⟨2, 5⟩ and w = ⟨6, 1⟩: v − w = ⟨2 − 6, 5 − 1⟩ = ⟨−4, 4⟩, and w − v = ⟨4, −4⟩, the additive inverse of the first answer. Subtraction is not commutative, exactly as with numbers.

Subtraction graphically: connecting the tips. Draw v and w from the same initial point. The vector v − w is the arrow connecting the tips in the appropriate order: from the tip of w to the tip of v. The order is the thing to remember: the arrow points toward the tip of the vector you are subtracting from. Why this works: starting at the tip of w, traveling along v − w must land you at the tip of v, because w + (v − w) = v. For v = ⟨2, 5⟩ and w = ⟨6, 1⟩ the tips are at (2, 5) and (6, 1), and the arrow from (6, 1) to (2, 5) is ⟨2 − 6, 5 − 1⟩ = ⟨−4, 4⟩, matching the component calculation. Drawing the arrow the wrong way round gives w − v. The other graphical method is to draw −w (reverse w) and add it end-to-end to v; both pictures give the same arrow.

The misconception. Students who have just learned that vectors can be added often add magnitudes: "a 3 N force plus a 4 N force is a 7 N force." That is true only if the two forces point the same way. At right angles the resultant is 5 N; opposite, 1 N; at any other angle, somewhere between 1 N and 7 N. Two vectors' magnitudes tell you the possible range of |u + w|, from ||u| − |w|| up to |u| + |w|; the angle between them decides where in that range the answer falls. A second error is subtracting in the graphical picture by drawing the arrow from the tip of v to the tip of w, which yields the inverse of the intended difference. A quick sign check on one component catches it.

Why all three additions agree. Placing w at the tip of u adds its horizontal move to u's horizontal move and its vertical move to u's vertical move; that is exactly what the component sum computes. The parallelogram is the same construction drawn twice, once with w first and once with u first, and both diagonals-from-the-origin meet at the same vertex, which is why vector addition is commutative: u + w = w + u.

Worked, end-to-end and component-wise: Add ⟨3, −2⟩ and ⟨−5, 4⟩. Component-wise: ⟨3 + (−5), −2 + 4⟩ = ⟨−2, 2⟩. End-to-end: from the origin go 3 right and 2 down to (3, −2), then 5 left and 4 up to (−2, 2); the sum arrow runs from (0, 0) to (−2, 2). Magnitudes: |⟨3, −2⟩| = √13 ≈ 3.606, |⟨−5, 4⟩| = √41 ≈ 6.403, sum of magnitudes ≈ 10.01, but |⟨−2, 2⟩| = √8 ≈ 2.828. The parts partly cancel because they point in nearly opposite directions.

Worked, magnitude and direction form: p has magnitude 12 at 90° and q has magnitude 5 at 0°. Components: p = ⟨0, 12⟩, q = ⟨5, 0⟩. Sum ⟨5, 12⟩, magnitude √(25 + 144) = 13, direction arctan(12/5) ≈ 67.38°. Check: 13 lies between 12 − 5 = 7 and 12 + 5 = 17, and 67.38° lies between 0° and 90°, closer to 90° because the 12-unit vector is larger.

Worked, equal magnitudes 60° apart: a = 10 at 0°, b = 10 at 60°. ⟨10, 0⟩ + ⟨5, 5√3⟩ = ⟨15, 5√3⟩ ≈ ⟨15, 8.660⟩; magnitude √300 = 10√3 ≈ 17.32 at exactly 30°, since tan θ = 5√3/15 = √3/3. Converting back: 17.32 cos 30° ≈ 15.0 and 17.32 sin 30° = 8.66. The sum bisects the angle because the two magnitudes are equal.

Worked, subtraction two ways: v = ⟨2, 5⟩, w = ⟨6, 1⟩. Component-wise: v − w = ⟨−4, 4⟩. Via the additive inverse: −w = ⟨−6, −1⟩, and v + (−w) = ⟨2 − 6, 5 − 1⟩ = ⟨−4, 4⟩. Graphically: draw both from the origin; the arrow from the tip of w at (6, 1) to the tip of v at (2, 5) is ⟨−4, 4⟩. |v − w| = √32 ≈ 5.657. Reversing the order gives w − v = ⟨4, −4⟩, the same length pointing the opposite way.

Worked, when magnitudes do add: ⟨2, 1⟩ + ⟨4, 2⟩ = ⟨6, 3⟩. Magnitudes: √5 ≈ 2.236, √20 ≈ 4.472, and √45 ≈ 6.708 = 2.236 + 4.472. The magnitudes add exactly because ⟨4, 2⟩ = 2⟨2, 1⟩ points in the same direction; the triangle has collapsed into a line.

Recording task: On grid paper draw u = ⟨4, 1⟩ and w = ⟨1, 3⟩ from the origin. Construct the sum three times: end-to-end (copy w to the tip of u), by the parallelogram rule (complete the parallelogram and draw the diagonal from the origin), and component-wise in a table. Confirm all three give ⟨5, 4⟩. Measure the three arrows with a ruler in grid units and record |u| ≈ 4.12, |w| ≈ 3.16 and |u + w| ≈ 6.40, then write the inequality 6.40 < 4.12 + 3.16 in words. Finally draw u − w by connecting the tip of w to the tip of u and check that it equals ⟨3, −2⟩ by components.

Independent performance: Choose two vectors of your own with different magnitudes, neither along an axis and not parallel. Present them first in component form and then, separately, in magnitude and direction form (magnitude to two decimals, angle to two decimals). Compute their sum by all three methods and their difference in both orders, both component-wise and graphically. Then design one pair of vectors for which the magnitude of the sum does equal the sum of the magnitudes and one pair for which the sum is the zero vector, and explain what is special about each. Success criteria: the three addition methods agree; the graphical difference arrow points from the tip of the subtracted vector to the tip of the other and matches the components; every magnitude is computed with the Pythagorean theorem and the inequality |u + w| ≤ |u| + |w| is stated numerically for each pair; and the magnitude-and-direction sum is converted back to components as a check.