How to calculate the area of a parallelogram built on vectors

On any two noncollinear and nonzero vectors, a parallelogram can be constructed. These two vectors will contract a parallelogram if you combine their origin at one point. Finish the sides of the figure.

Instruction manual

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Find the lengths of the vectors if their coordinates are given. Let, for example, the vector A have coordinates (a1, a2) in the plane. Then the length of the vector A is | A | = √ (a1² + a2²). Similarly, we find the module of the vector B: | B | = √ (b1² + b2²), where b1 and b2 are the coordinates of the vector B on the plane.

The parallelogram area is found by the formula S = | A | • | B | • sin (A ^ B), where A ^ B is the angle between the given vectors A and B. The sine can be found through the cosine using the basic trigonometric identity: sin²α + cos²α = one. The cosine can be expressed in terms of the scalar product of vectors written in coordinates.

The scalar product of a vector A by a vector B is denoted by (A, B). By definition, it is equal to (A, B) = | A | • | B | • cos (A ^ B). And in coordinates, the scalar product is written like this: (A, B) = a1 • b1 + a2 • b2. From here we can express the cosine of the angle between the vectors: cos (A ^ B) = (A, B) / | A | • | B | = (a1 • b1 + a2 • b2) / √ (a1² + a2²) • √ (a2² + b2²). In the numerator, the scalar product; in the denominator, the lengths of the vectors.

Now we can express the sine from the main trigonometric identity: sin²α = 1-cos²α, sinα = ± √ (1-cos²α). If we assume that the angle α between the vectors is acute, the minus with the sine can be discarded, leaving only the plus sign, since the sine of the acute angle can only be positive (or zero at zero angle, but here the angle is non-zero, this is displayed in the condition noncollinearity of vectors).

Now we need to substitute the coordinate expression for the cosine in the sine formula. After this, it remains only to write the result in the parallelogram area formula. If all this is done and the numerical expression is simplified, then it turns out that S = a1 • b2-a2 • b1. Thus, the area of the parallelogram constructed on the vectors A (a1, a2) and B (b1, b2) is found by the formula S = a1 • b2-a2 • b1.

The resulting expression is the determinant of the matrix composed of the coordinates of the vectors A and B: a1 a2b1 b2.

Indeed, in order to obtain a determinant of a matrix of dimension two, we need to multiply the elements of the main diagonal (a1, b2) and subtract from this the product of the elements of the side diagonal (a2, b1).