The vector product: definition, right-hand rule and the products of i, j, k
Pure Mathematics Sec 3 Term 1, Unit 3, Lesson 3
GoalDefine the vector product A × B, find its size and its direction by the right-hand rule, and use the products of i, j and k.
New words
vector product · A × B = (‖A‖‖B‖ sin θ) c: a vector perpendicular to both A and Bright-hand rule · fingers of the right hand turn from A to B through the angle θ; the thumb points along A × Bzero vector · the vector of size 0; A × B is the zero vector when A and B are parallel
Explanation
Last week the scalar product of two vectors gave a number. The second way of multiplying two vectors gives a vector, and it is called the vector product, or cross product (in Arabic الضرب الاتجاهي). Draw A and B from the same point, with θ between them (0° ≤ θ ≤ 180°). Then A × B is the vector with these two properties. Its size is ‖A × B‖ = ‖A‖‖B‖ sin θ. Its direction is perpendicular to the plane that holds A and B, and the sense is given by the right-hand rule: curl the fingers of your right hand from A towards B through the angle θ, and the thumb points along A × B. In symbols A × B = (‖A‖‖B‖ sin θ) c, where c is the unit vector along the thumb. Three facts follow. First, the order matters: turning from B to A reverses the thumb, so B × A = −(A × B). Second, if A and B are parallel then sin θ = 0 (θ = 0° or 180°), so A × B is the zero vector; in particular A × A is the zero vector. Third, if A and B are perpendicular then sin θ = 1 and ‖A × B‖ = ‖A‖‖B‖, the largest value. For the unit vectors of the axes the right-hand rule gives i × j = k, j × k = i and k × i = j, and the reverse orders give the negatives: j × i = −k, k × j = −i and i × k = −j. Compare the two products: A · B uses cos θ and is a number; A × B uses sin θ and is a vector.
row × column: i × j = k, but j × i = −k; 0 is the zero vector
×
i
j
k
i
0
k
−j
j
−k
0
i
k
j
−i
0
Worked examples
Let ‖A‖ = 8 and ‖B‖ = 5. For θ = 30° the size is 40 sin 30° = 20. For θ = 90° it is 40. For θ = 150° it is 40 sin 150° = 20, the same as for 30°, because sin 150° = sin 30°. For θ = 0° it is 0.
Products of unit vectors: (2i) × (3j) = 6(i × j) = 6k. Also j × (4i) = 4(j × i) = −4k, (3k) × i = 3(k × i) = 3j, and k × (−2j) = −2(k × j) = 2i.
If ‖A‖ = 6, ‖B‖ = 4 and ‖A × B‖ = 12, then 24 sin θ = 12, so sin θ = 1/2 and θ = 30° or θ = 150°. Both angles are possible: the sine alone cannot tell them apart. If we are also told that A · B is positive, then the angle is acute and θ = 30°.
Both products together: if ‖A‖ = 3, ‖B‖ = 5 and A · B = 9, then cos θ = 9/15 = 3/5, so sin θ = 4/5 (a 3-4-5 triangle) and ‖A × B‖ = 15 × 4/5 = 12.
Class activity
The right-hand test. Hold your right hand with the fingers pointing along the x-axis of the room (say, towards the door) and curl them towards the y-axis (say, towards the window). Where does the thumb point? Check that it is the z-axis, i × j = k. Now curl from the y-axis to the x-axis and see the thumb turn round: j × i = −k.
Practice
1
Find ‖A × B‖ when ‖A‖ = 7, ‖B‖ = 4 and θ = 30°.
7 × 4 × sin 30° = 28 × 1/2 = 14.
2
Simplify (4i) × (3j), (2j) × (5k), (3k) × i and j × (−2i).
