Add arrows and find the actual resultant

Try

A swimmer moves at (3,0) metres per second relative to still water. A current moves at (0,4) metres per second relative to the bank. Under a constant-velocity model in the same east/north frame, the swimmer’s velocity relative to the bank is (3,4) metres per second. Its speed is 5, not 7.

The two speeds cannot simply be added because the motions point in different directions. Construct the actual resultant arrow before calculating its magnitude. This fixed example is separate from the adjustable inputs below.

Predict

Predict what changes if the current becomes (0,−4). The resultant speed remains 5, but the direction reflects across the horizontal axis. If the current instead becomes (−3,0), the resultant is zero and has no direction.

For subtraction u−v, predict the correct tip-to-tip arrow when u and v begin at the same origin. Starting at the tip of v and ending at the tip of u gives u−v. Reversing that arrow gives the negative answer.

Experiment

Use the live construction to compare addition, subtraction and signed scaling. Its controls describe velocity components in metres per second, on fixed equal axis scales. It is a separate playground, not the fixed swimmer calculation. Choose one pair, predict the result and its magnitude, then inspect both the arrows and numerical summary.

Build the resultant, then explain its direction

Think of u and v as velocities measured in metres per second. Before changing a component, predict how the resultant will move. Addition combines velocities, subtraction compares them, and the dimensionless factor c rescales one velocity. Speed is a magnitude; velocity also has direction.

Operationu + vu − vc × u u horizontal-2-1012u vertical-2-1012 v horizontal-2-1012v vertical-2-1012 Reset vectors

Components → speed → direction

r = (1, 3) m/s

Squared magnitude: (1)² + (3)² = 10. Speed = √10 ≈ 3.162 m/s.

Direction ≈ 71.565° counterclockwise from the positive horizontal axis. Both components determine the quadrant.

u = (2, 1) m/s; v = (-1, 2) m/s.

The dashed arrow copies v with its tail at the tip of u. Its endpoint is the tip of r. Translation changes the location of an arrow, not its vector components.

Arrows can overlap; use the labeled component summary when they do. The axes never rescale. Angles and square roots are rounded displays. Try a cancellation yielding zero, then a negative scalar. Settings are temporary; justify a separate construction in the lesson practice.

For addition, trace the second arrow translated to the first tip. For subtraction, trace the arrow from the second tip to the first. Set equal vectors and subtract them; the zero result must have magnitude zero and no direction. Next make opposite vectors and add them. Explain why these two different operations can lead to the same zero result.

Fixed construction	First tip	Second translated tip	Result
(3,0)+(0,4)	(3,0)	(3,4)	(3,4)
(3,0)−(0,4)	(3,0)	(3,−4)	(3,−4)

The table uses the fixed example, not the current widget values.

Understand

For u=(a,b) and v=(c,d), translating v to the tip of u reaches (a+c,b+d), which defines u+v. Starting with v instead reaches the same point because real addition commutes. The four points (0,0),u,u+v,v form a parallelogram when the vectors are not parallel; its diagonal from the origin is the sum. Parallel or zero cases may collapse the parallelogram, while component addition remains valid.

The opposite vector is −v=(−c,−d). Adding it gives u−v=(a−c,b−d). These are also the coordinates of the displacement from v’s tip to u’s tip. Thus the end-to-end and tip-to-tip subtraction constructions agree exactly, not just visually.

For magnitude-direction input, first convert each vector to components. Two length-4 vectors at 0° and 120° have components (4,0) and (−2,2√3). Their sum is (2,2√3), magnitude 4 and direction 60°. The components, not the angles, are added. Adding angles would describe a different operation.

After computing a nonzero resultant (x,y), its length is √(x²+y²). Recover direction from the signs and trigonometry. An ordinary inverse tangent gives a restricted angle and can miss the quadrant; for a negative x and positive y, use a quadrant-II angle. If x=0 use the vertical direction, and if both components vanish report direction as undefined.

Apply

In a drawing response, identify both original arrows, their translated copy and the resultant. Exact coordinates provide an alternative to a physical drawing, but must include directed segments and the chosen axis convention.

Use a fresh velocity scenario in practice and state what is being added relative to what. Two unrelated velocities do not become a meaningful combined physical velocity merely because their units match. The water-relative plus bank-relative example supplies the relationship that justifies addition.

Continue

These constructions address all three parts of N-VM.B.4: component and geometric addition, magnitude-direction inputs, and subtraction with the correct arrow orientation.

Next, investigate scalar multiplication. A negative scalar reverses a nonzero vector, a positive scalar preserves its direction, and a zero scalar removes directional information. The same widget can help test those statements, but the general magnitude formula needs an argument using components.