MCV4U The Cross Product: Notes and Practice

Loosening a stuck bolt with a wrench depends on three things: how hard you push, how far from the bolt you push, and the angle between your push and the handle. Push along the handle and nothing turns; push square to it at the very end and you get the most turning effect. Notice too that the bolt turns about an axis that is perpendicular to both the handle and your push. A product that takes two vectors and returns a vector perpendicular to both, with a size that grows with $sin θ$ , is exactly what this situation needs. That is the CROSS PRODUCT, and in this course it is defined only for vectors in 3-space.

The definition

a \times b = (a_{2} b_{3} - a_{3} b_{2}, a_{3} b_{1} - a_{1} b_{3}, a_{1} b_{2} - a_{2} b_{1})

For a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃). The result is a vector.

A reliable way to remember it: write the components of $a$ in a row, then those of $b$ underneath. To get the first component, cover the first column and cross-multiply what is left: $a_{2} b_{3} - a_{3} b_{2}$ . For the second component, cover the second column but cross-multiply in the order that starts from the THIRD column, $a_{3} b_{1} - a_{1} b_{3}$ (this is the one people get backwards). For the third, cover the third column: $a_{1} b_{2} - a_{2} b_{1}$ . Always check your answer by dotting it with $a$ and with $b$ : both results must be $0$ .

Fig 1Vectors a and b in a plane, with a × b perpendicular to both by the right-hand rule, and the parallelogram whose area is |a × b| = |a||b|sin θ. Reversing the order gives b × a = −(a × b).

Direction: $a \times b$ is perpendicular to both $a$ and $b$ . Point the fingers of your right hand along $a$ and curl them toward $b$ ; your thumb points along $a \times b$ (the RIGHT-HAND RULE).
Magnitude: $∣ a \times b ∣ = ∣ a ∣∣ b ∣ sin θ$ , where $0° \leq θ \leq 180°$ is the angle between the vectors tail to tail.
Unit vectors: $i \times j = k$ , $j \times k = i$ , $k \times i = j$ , and reversing any pair gives the negative.

VideoMr. Cioffi's MCV4U lesson on the cross product of two vectors in 3-space.Watch on YouTube

Properties

Property	Statement
Anti-commutative	$b \times a = - (a \times b)$
Collinear vectors	$a \times a = 0$ , and $a \times (k a) = 0$ since $sin 0° = sin 180° = 0$
Distributive	$a \times (b + c) = a \times b + a \times c$
Scalars factor out	$(k a) \times (m b) = km (a \times b)$
Perpendicular to both	$a \cdot (a \times b) = 0$ and $b \cdot (a \times b) = 0$
Not associative	$(i \times i) \times j = 0$ , but $i \times (i \times j) = i \times k = - j$

Unlike ordinary multiplication, order and grouping both matter for the cross product.

Area of a parallelogram and a triangle

A parallelogram with sides

a

and

b

has base

∣ a ∣

and height

∣ b ∣ sin θ

, so its area is

∣ a ∣∣ b ∣ sin θ = ∣ a \times b ∣

. A triangle with those two sides is half of it: area

= \frac{1}{2} ∣ a \times b ∣

. For a triangle given by three points

A

B

C

, use

a = A B

and

b = A C

Torque

Fig 2A wrench of length r with a force F applied at angle θ to the handle. The torque magnitude is |r × F| = |r||F|sin θ, largest when θ = 90 degrees.

The turning effect of a force is the TORQUE, $τ = r \times F$ , where $r$ runs from the pivot (the bolt) to the point where the force acts. Its magnitude is $∣ r ∣∣ F ∣ sin θ$ newton-metres, and its direction (by the right-hand rule) lies along the axis the bolt turns about. To maximize torque, push at the far end of the handle (large $∣ r ∣$ ) and at $90°$ to it ( $sin θ = 1$ ). In a cramped engine bay you often cannot swing the handle far. A ratchet lets the wrench turn the bolt one way and click freely back the other, so you can make short strokes and reset your grip each time, keeping the push close to $90°$ instead of lifting the wrench off and refitting it.

Close-up of a beam-type torque wrench with a curved scale marked in inch-pounds on one side and newton-metres on the other, and a red pointer. — PhotoA beam-type torque wrench shows how hard a bolt is being turned, with one scale in newton-metres. That reading is the magnitude of the torque described in this section.EncMstr, CC BY-SA 3.0, via Wikimedia Commons

VideoApplies the cross product to torque, finding the turning effect of a force applied at a distance from a pivot.Watch on YouTube

Volume of a parallelepiped

A parallelepiped is a slanted box whose edges from one corner are $a$ , $b$ and $c$ . Its base, the parallelogram on $b$ and $c$ , has area $∣ b \times c ∣$ , and $b \times c$ is perpendicular to that base. The height is the scalar projection of $a$ onto $b \times c$ . Multiplying base by height leaves $V = ∣ a \cdot (b \times c) ∣$ , the absolute value of the SCALAR TRIPLE PRODUCT. If it equals $0$ , the box is flat: the three vectors lie in one plane (they are coplanar).

The Cross Product

Key things to remember

The definition

Properties

Torque

Volume of a parallelepiped

Worked examples

Where marks get lost

Flashcards 9

Practice