C. Geometry and Algebra of Vectors · Dot and Cross Products

Solving Problems with Dot and Cross Products

C2.8 C2.4 C2.6About 75 minutes

Learning goals

I can choose the right vector tool (dot product, projection, cross product or triple scalar product) for a problem.
I can solve ground-velocity, headwind and crosswind problems using vectors and the dot product.
I can find normals, areas and angles for triangles and planes in three-space.
I can combine several vector operations in one multi-step problem and check the result.

Words to know

heading
The direction an aircraft or boat points (its velocity relative to the air or water).

track
The actual path over the ground, in the direction of the ground velocity.

ground velocity
Velocity relative to the ground: air (or water) velocity plus wind (or current) velocity.

headwind / crosswind
The components of the wind against (or along) the direction of travel, and across it.

Choosing the right tool

You now have a full vector toolbox. The skill in this lesson is choosing the right tool for a problem, often using two or three in a row. Most problems start with a sketch, turn the information into vectors (in components if possible), and then apply one of the tools in the table below.

This lesson also brings back the airplane and wind problems from Vectors in Two-Space and Three-Space. There you found a ground velocity by adding vectors. Now the dot and cross products let you answer more questions about the same situation.

You want…	Use…	Lesson
the angle between two directions	cos θ = (u · v)/(\|u\|\|v\|)	4
to test for a right angle	u · v = 0	3
the part of a vector along a direction	scalar or vector projection	5
work done by a force	W = F · d	6
a vector perpendicular to two vectors (a normal)	u × v	7
the area of a parallelogram or triangle	\|u × v\| or ½\|u × v\|	8
to test for parallel vectors	u × v = 0	8
torque	τ = r × F	10
volume, or a test for coplanar vectors	\|a · (b × c)\|	11

The vector toolbox for this unit.

The plane points north at 300 km/h relative to the air, but the wind pushes it east and slows it slightly. The ground velocity g = a + w; the dot product gives the angle θ between the heading and the actual track.

Worked example

Example 1: Ground velocity and drift

An airplane heads due north with an airspeed of 300 km/h. The wind velocity is w = [60, −20] km/h (x east, y north). Find the ground speed and the angle between the plane's heading and its actual track.

Air velocity: a = [0, 300]. Ground velocity: g = a + w = [60, 280].
Ground speed: |g| = √(60² + 280²) = √82 000 ≈ 286.4 km/h.
Angle between heading and track: cos θ = (a · g)/(|a||g|) = (0 + 84 000)/(300 × 286.36) ≈ 0.9778, so θ ≈ 12.1°.
Check: tan θ = 60/280 gives θ ≈ 12.1° as well ✓.

Ground speed about 286 km/h; the track is about 12.1° east of north (a bearing of about 012°). The wind blows the plane east and slightly slows it.

A wind w = [−10, −30] km/h splits into a part along the runway (a 30 km/h headwind for a plane moving in direction d) and a 10 km/h crosswind. The parts are a projection and what is left over.

Worked example

Example 2: Headwind and crosswind

A runway points in the direction d = [3, 4]. The wind velocity is w = [−10, −30] km/h. For a plane taking off in the direction of d, find the headwind (the wind component along the runway) and the crosswind (the component across it).

Scalar projection of w on d: (w · d)/|d| = (−30 − 120)/5 = −30. The wind blows 30 km/h against the direction of travel: a 30 km/h headwind.
|w|² = 100 + 900 = 1000. The crosswind is the perpendicular part: √(1000 − 30²) = √100 = 10 km/h.
Check with vectors: the projection is (−150/25)[3, 4] = [−18, −24], and w − [−18, −24] = [8, −6], which has length 10 and is perpendicular to d ([8, −6] · [3, 4] = 0) ✓.

30 km/h headwind and 10 km/h crosswind. Pilots prefer to take off into a headwind, because it shortens the takeoff run.

Worked example

Example 3: A normal, an area and a point test

A(1, 0, 0), B(0, 2, 0) and C(0, 0, 3) are points. (a) Find a vector perpendicular to the triangle ABC. (b) Find the area of the triangle. (c) Does D(0, 1, 1.5) lie in the same plane as A, B and C?

(a) AB = [−1, 2, 0], AC = [−1, 0, 3]. n = AB × AC = [6, 3, 2]. Check: n · AB = −6 + 6 = 0 ✓, n · AC = −6 + 6 = 0 ✓.
(b) Area = ½|n| = ½√(36 + 9 + 4) = ½(7) = 3.5.
(c) AD = [−1, 1, 1.5]. If D is in the plane, AD lies in the plane and must be perpendicular to n: n · AD = −6 + 3 + 3 = 0.

(a) [6, 3, 2] (or any non-zero multiple); (b) 3.5 square units; (c) yes, D is in the plane. In Lines and Planes you will use exactly this idea: a normal vector and one point describe a whole plane.

Worked example

Example 4: A tilted solar panel

A triangular solar panel has corners O(0, 0, 0), A(4, 0, 0) and B(0, 3, 2), in metres, with the ground as the xy-plane. Find its area and the angle it makes with the ground.

OA × OB = [(0)(2) − (0)(3), (0)(0) − (4)(2), (4)(3) − (0)(0)] = [0, −8, 12].
Area = ½√(0 + 64 + 144) = ½√208 ≈ 7.21 m².
The angle between two planes equals the angle between their normals. The ground's normal is k = [0, 0, 1].
cos θ = ([0, −8, 12] · [0, 0, 1])/(√208 × 1) = 12/√208 ≈ 0.8321, so θ ≈ 33.7°.

Area about 7.21 m², tilted about 33.7° from the ground. You could find the tilt of a real installation the same way, from measured corner positions.

Worked example

Example 5: Work along a path

A constant force F = [10, −4, 6] N pushes a cart from A(1, 2, 0) to B(4, 6, 1), then on to C(4, 6, 5) (metres). Find the total work.

AB = [3, 4, 1] and BC = [0, 0, 4].
W₁ = F · AB = 30 − 16 + 6 = 20 J and W₂ = F · BC = 0 + 0 + 24 = 24 J.
Total 44 J. Shortcut: F · AC = [10, −4, 6] · [3, 4, 5] = 30 − 16 + 30 = 44 J ✓ (the distributive property again).

44 J. For a constant force, only the overall displacement matters, not the path.

Just for fun

Pilots talk about headwinds and crosswinds all day without ever saying "scalar projection". Now you know what their numbers really are, and you can say it at the airport to confuse your relatives.

Hands-on activity

A toy car crossing a moving river

You need: a toy car (or a coin), a large sheet of paper (or a tea towel) on a smooth table, a ruler, a pencil, a watch or phone timer and a helper.

Draw a straight line across the paper from one long edge to the other. This is where the car aims (its heading).
Your helper slowly and steadily pulls the paper sideways (the "river current") while you push the car along the drawn line at a steady speed. Time the crossing.
Mark where the car started and finished on the table (use small pieces of tape on the table). The line between them is the ground track.
Measure: the car's speed along the paper (line length ÷ time), the paper's speed (how far it moved ÷ time) and the ground track. Write the velocities as vectors and check that ground velocity = car velocity + paper velocity.
Use the dot product to find the angle between the heading and the track, and compare with a protractor measurement.

What to look for: for example, if the car goes 30 cm across in 6 s (5 cm/s) while the paper moves 12 cm sideways (2 cm/s), the ground velocity is [2, 5] cm/s, the ground speed is about 5.4 cm/s and the track is about 21.8° from the heading, since cos θ = 5/√29 ≈ 0.928.

Your turn

Which tool would you use for each: (a) the angle between two forces, (b) the area of a triangular sail, (c) the turning effect of a force on a lever, (d) whether three vectors lie in one plane?

A plane has air velocity [0, 400] km/h and the wind is [50, 0] km/h. Find the ground speed and the angle between heading and track.

F = [2, 3, −1] N acts through d = [4, 0, 2] m. Find the work.

Find the area of the triangle with vertices A(0, 0, 1), B(2, 0, 1) and C(0, 4, 3).

Find a normal vector to the plane containing the vectors [1, 0, 2] and [0, 1, −1].

The wind is w = [20, −15] km/h and a runway points along [1, 0]. For a plane moving in the direction [1, 0], find the headwind or tailwind and the crosswind.

More practice: warm-up

Dot or cross: work done by a force?

Dot or cross: torque?

Dot or cross: area of a parallelogram?

Dot or cross: the angle between two vectors?

Which product gives the volume of a parallelepiped?

More practice: on your own

A plane heads on a bearing of 060° with an airspeed of 250 km/h. The wind blows from the north at 40 km/h. Find the ground speed and the bearing of the track.

A runway points northwest, d = [−1, 1]. The wind is w = [30, 10] km/h. Find the headwind and crosswind for a plane moving along d.

For A(2, 0, 0), B(0, 2, 0) and C(0, 0, 2), find ∠A and the area of △ABC.

A 2 kg ball (weight [0, 0, −19.6] N) moves from (0, 0, 5) to (3, 4, 0), in metres. Find the work done by gravity.

A force F = [0, 0, −50] N acts at r = [0.3, 0.4, 0] m from a pivot. Find τ and |τ|.

Challenge

Find the distance from P(1, 2, 3) to the line through the origin with direction d = [1, 1, 0]. (Hint: the parallelogram on OP and d has area |OP × d|; divide by the base |d|.)

A rectangular solar panel has edges [2, 0, 0] and [0, 1.2, 0.9] (metres) from one corner. Find its area and its tilt from the ground.

A pilot wants to fly due east with an airspeed of 200 km/h, and the wind is [0, 30] km/h (blowing north). What heading should the pilot fly, and what is the ground speed?

Check your understanding

Quiz

Which tool finds the area of a triangle with vertices given in three-space?

Which test shows that two non-zero vectors are perpendicular?

A plane has air velocity [0, 200] km/h and the wind is [30, 0] km/h. What is its ground speed?

Which vector is perpendicular to the plane containing u and v?

The wind is [12, 5] km/h and a runway points along [1, 0]. What is the crosswind?

A force F = [1, 2, 3] N moves an object through d = [3, 2, 1] m. How much work is done?

For parents and teachers

Big idea (C2.8, with C2.4 and C2.6): real problems often need several vector tools in a row: add vectors for a ground velocity, then use the dot product for an angle; use a cross product for a normal, then a dot product to test a point. The table at the start of the lesson summarizes which tool does which job.

How to help: Ask your teen to explain each row of the toolbox table in their own words. The toy-car activity is a fun way to see that a heading and a track are different vectors.

Common difficulties:

Choosing the cross product for an angle (use the dot product) or the dot product for an area (use the cross product).
Mixing up "heading" (where the plane points) and "track" (where it actually goes).
Bearing conventions: bearings are measured clockwise from north, so with x east and y north, a bearing of β gives the direction [sin β, cos β].
Forgetting to check a normal vector by dotting it with both original vectors.