Cross product or vector product of two vectors:

It is given by the following formula:

where n^ is unit vector which is perpendicular to plane containing the vectors P and Q and its direction is governed by right hand screw rule.

The cross product of two vectors P and Q is also given by the following formula:

where i, j, and k are unit vectors along +x, +y, and +z axes. Px, Py, and Pz are x, y and z components of the vector P. Qx, Qy, and Qz are x, y, and z components of the vector Q.

You can compute the cross product of P and Q by putting the values of components of P and Q in the following formula:

P × Q = i ((Py × Qz) - (Pz × Qy)) + j ((Pz × Qx) - (Px × Qz))
+ k ((Px × Qy) - (Py × Qx))

Direction of Cross product of two vectors:

If R = P x Q (i.e., R is the result of cross product of P and Q) then direction of R is given by right hand rule. Stretch the forefinger, middle finger and thumb of your right hand mutually perpendicular to each other. Now, if forefinger gives the direction of P and middle finger gives the direction of Q then thumb gives the direction of R. If P and Q are not perpendicular to each other then pick the component of P that is perpendicular to Q and use it for alignment along forefinger.

Example 1 of cross product of two vectors:

Magnitude of vector P is 20 units, magnitude of vector Q is 40 units, and angle θ between them is 30^o. Find the magnitude of cross product of vectors P and Q.

Solution:

Let vector R represents the cross product of P and Q as follows:

R = P × Q

Now magnitude of R is given by the formula:

R = P × Q × sinθ

Put values in this formula and we get:

R = P × Q × sinθ = 20 units × 40 units × sin(30^o)

R = 20 units × 40 units × 0.5000 = 400 units

R = 400 units -------------------------(Answer)

Example 2 of cross product of two vectors:

Let P = 3 i - 4 j + 7 k and let Q = -5 i + 9 j + 6 k. Find the cross product of P and Q.

We can find the cross product of P and Q using the following formula:

P × Q = i ((Py × Qz) - (Pz × Qy)) + j ((Pz × Qx) - (Px × Qz))
+ k ((Px × Qy) - (Py × Qx))

Here, the values of components of P and Q are as follows:

Px = 3, Py = -4, Pz = 7, Qx = -5, Qy = 9, and Qz = 6.

Let R = P x Q (i.e., vector R represents the cross product of vectors P and Q), then components of R, namely, Rx, Ry, and Rz are given by:

Rx = (Py × Qz) - (Pz × Qy)

Ry = (Pz × Qx) - (Px × Qz)

Rz = (Px × Qy) - (Py × Qx)

Put the values in these expressions and we get:

Rx = (Py × Qz) - (Pz × Qy) = (-4 × 6) - (7 × 9) = -87

Ry = (Pz × Qx) - (Px × Qz) = (7 × -5) - (3 × 6) = -53

Rz = (Px × Qy) - (Py × Qx) = (3 × 9) - (-4 × -5) = 7

Therefore, we get:

Rx = -87

Ry = -53

Rz = 7

Therefore,

R = Rx i + Ry j + Rz k = -87 i - 53 j + 7 k

R = -87 i - 53 j + 7 k --------------------(Answer)

About Shirish Chavan

CEO of Webilog, Mr. Shirish Chavan, is an internationally acknowledged computer programmer. He has authored more than ten books on computers and some of them were bestsellers. All the study material available on this website is written by him with great care.

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