A vector is a mathematical construct that has magnitude and direction. The cross product of two vectors is a binary operation in three-dimensional space that results in a third vector that is perpendicular to the plane that contains the two input vectors. The direction of the resulting vector is determined by the order of the input vectors, so the vector cross product does not have an associative or commutative property. The vector cross product has extensive uses in mathematics and physics, as well as practical applications in computer graphics.

### Step 1

Use the right-hand rule to determine the direction of the resulting vector in a cross product. Hold your right hand in front of you so that the thumb is pointed up, the index finger is pointed away from you and the middle finger is pointed to your left. The index finger shows the direction of vector A, the middle finger shows the direction of vector B and the thumb shows the direction of the vector from the cross product A x B.

### Step 2

Define the cross product as A x B = ab sin θ n, where A and B are the input vectors, a is the magnitude of vector A, b is the magnitude of vector B, θ is the smaller angle's measure between A and B, and n is a unit vector perpendicular to the plane that contains vectors A and B. The direction of n is given by the right-hand rule in Step 1.

### Step 3

Define some notation. Let i, j and k be the unit vectors in a given orthogonal coordinate system. We can now say A = a1i + a2j + a3k = (a1, a2, a3), where a1 is the magnitude of i, a2 is the magnitude of j and a3 is the magnitude of k. Similarly, B = b1i + b2j + b3k = (b1, b2, b3). We must also establish the following identities for unit vectors: i x j = k, j x k = i, k x i = j, j x i = 'k, k x j = 'i, i --- k = 'j, i x i = 0, j x j = 0, k x k = 0.

### Step 4

Use distributive cross multiplication to calculate the cross product. A x B = (a1i + a2j + a3k) --- (b1i + b2j + b3k) = a1i --- (b1i + b2j + b3k) + a2j --- (b1i + b2j + b3k) + a3k --- (b1i + b2j + b3k) = (a1i --- b1i) + (a1i --- b2j) + (a1i --- b3k) + (a2j --- b1i) + (a2j --- b2j) + (a2j --- b3k) + (a3k --- b1i) + (a3k --- b2j) + (a3k --- b3k).

### Step 5

Factor out the unit vectors from the result obtained in Step 4 to obtain A x B = a1b1(i x i) + a1b2(i x j) + a1b3(i x k) + a2b1(j x i) + a2b2(j x j) + a2b3(j x k) + a3b1(k x i) + a3b2(k x j) + a3b3(k x k). Now apply the identities for unit vectors given in Step 3 to obtain the following: A x B = a1b1(0) + a1b2(k) + a1b3('j) + a2b1('k) + a2b2(0) + a2b3(i) + a3b1(j) + a3b2('i) + a3b3(0) = a1b2(k) ' a1b3(j) ' a2b1(k) + a2b3(i) + a3b1(j) ' a3b2(i).

### Step 6

Collect the coefficients of each vector in the result of Step 5 to obtain A x B = (a2b3 ' a3b2) i + (a3b1 ' a1b3) j + (a1b2 ' a2b1) k. Using the notation given in Step 3, we may now say A x B = (a2b3 ' a3b2, a3b1 ' a1b3, a1b2 ' a2b1).

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