A controller describes an arrow by the vector v = (3, 4). It needs a second arrow pointing in exactly the same direction, but with magnitude 1.

Find the new vector. Explain why your operation changes its length without changing its direction.

Now consider w = (−6, 8). Apply the same idea without finding an angle. Write a rule that works for any non-zero vector (a, b).

After recording your answer, select Show unit vector to compare the first pair.

For an acute angle θ, your unit vector has components cos θ and sin θ. The familiar triangle ratios now give the coordinates directly because its length is 1.

A unit circle has centre (0, 0) and radius 1. We extend the meaning of sine and cosine to every directed angle by defining

Measure θ from the positive x-axis: anticlockwise is positive; clockwise is negative. A coordinate is signed even though a triangle side length is not.

Move P beyond π/2. Explain what the negative horizontal coordinate means.

A unit vector starts at (1, 0). Three instructions rotate it through π/3, 7π/3 and −5π/3 radians.

Predict whether the final arrows coincide. Explain how the journeys differ.

On a circle of radius 1, one radian sweeps an arc of length 1. A full turn is 2π radians, so adding 2π changes the turn count but not the final direction.

Enter each instruction in the radians box. You may type pi/3, 7pi/3 or −5pi/3.

A sensor reports the vertical coordinate of a point on the unit circle as exactly 0.6. It does not report the horizontal coordinate.

Construct every possible point. Explain why your list is complete.

Call the acute direction α. Express the other direction in terms of α without using inverse trigonometry. Then state what extra information would distinguish the two points.

Use the circle to test your predictions. The dashed line marks y = 0.6; it is not a second circle.

Let the pinned unit vector be P = (a, b) at an acute angle α. Predict the image of P under each transformation before pressing its button.

Write each new coordinate pair and its directed angle. Which moves swap the components?

Every button acts on the purple pinned starting point, not on the last image. You can move P and pin a different starting point to test a general rule.

The line through O and a unit-circle point P = (x, y) has gradient y/x, provided x ≠ 0. Substituting the coordinate definitions gives

Find two different unit directions whose gradient is −3/4. Does knowing tangent identify one direction?

Sketch both arrows before using Opposite direction. Explain why a half-turn leaves the gradient unchanged even though the vector reverses.

Compare directions just before π/2, exactly at π/2 and just after π/2. Predict the gradient before selecting each position.

Explain why tangent is undefined at π/2, rather than zero or an ordinary very large number.

The vertical coordinate approaches 1 while the horizontal coordinate approaches 0. The sign of that small horizontal coordinate changes as the arrow passes the vertical.

Give every vertical direction using π and an integer k.

A rotating arrow has length 1. A programmer makes these statements about its direction.

Which statement is always true? Give a diagram or counterexample for the other two.

A graphing window shows only part of a function. Use the rotating vector to explain the whole functions, including negative angles and repeated turns.

Why are sine and cosine defined for every real angle, but restricted to values from −1 to 1?

For each function, state the period and identify the zeros and extrema over 0 ≤ θ ≤ 2π. Explain at least one feature using a point on the circle, not a memorised graph.

At a particular angle θ, a rotating unit vector has sin θ = −0.96 and points to the right of the y-axis. The readings are exact.

Find its horizontal coordinate. Predict the sine and cosine readings after an anticlockwise quarter-turn.

Explain how your predictions would appear on the two coordinate graphs. Use the vector transformation rather than calculating θ.

Every point P = (x, y) on the unit circle has distance 1 from O, regardless of the signs of its coordinates.

Write a relationship between x and y that rotation cannot change. Translate it into sine and cosine.

Justify the relationship for every point, including points on the axes. A table of a few correct readings is evidence, but it does not explain every possible angle.

After writing your argument, select Check magnitude and move the point to test it.

A unit vector points into quadrant III, so π < θ < 3π/2. Its vertical component is sin θ = −5/13.

Find cos θ and tan θ. Then find the vector of magnitude 26 in this same direction.

Show the squared quantity before choosing a sign. Explain how the interval for θ removes the ambiguity created when you take a square root.

A device reports the exact unit-direction components cos θ = −0.7 and sin θ = −0.8 for the same angle.

Can both readings be correct? Give a test that does not require finding θ.

A second device reports the vector (−7, −8) but does not claim it has length 1. Explain why that vector is valid, and how you would recover its unit direction.

A rotation through θ about O sends e₁ = (1, 0) to (cos θ, sin θ). The second basis vector, e₂ = (0, 1), begins a quarter-turn anticlockwise from e₁.

Where must e₂ go under the same rotation? Use a coordinate sketch before selecting “Show second image”.

The two transformed arrows must keep length 1, stay perpendicular and preserve their anticlockwise order. Test your prediction by changing θ.

The columns of a transformation matrix are the images of e₁ and e₂. Place the two vectors from your diagram in the correct columns.

Explain why each column has this form, then multiply Rθ by a general vector (x, y).

Use column vectors for the matrix multiplication. The bracketed matrix acts on any vector, not only a unit vector.

A controller uses the matrix

It acts on the vector (5, 0). All entries are exact.

Find the image and verify its length. Identify the rotation angle in 0 ≤ θ < 2π, to three significant figures.

Explain which matrix entries give cosine and sine. Do not use an inverse tangent result without checking its direction.

A few successful numerical tests do not establish that a matrix preserves every vector’s length. Use the general image

Show that x′² + y′² = x² + y².

Expand the two squares. Identify which terms cancel and where the unit-circle identity is needed. Then explain the result in a sentence.

You previously met the idea that multiplying by the imaginary number i rotates a point through π/2 anticlockwise. Its coordinate matrix is

Starting from (1,0), predict the position after one, two, three and four applications of J.

Use Apply J to check each prediction. Explain why two applications give the negative of the starting vector, while four restore it.

A vector of magnitude r > 0 points at directed angle θ. Start from (1,0), rotate through θ, then scale by r.

Would scaling first and rotating second give the same result? Explain using components or matrices.

A vector of magnitude 5 points in the direction of (−3,4). After a quarter-turn, state its coordinates, its magnitude and the change in direction.