Vector Functions and Operators

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Vector Functions and Operators The functions and overloaded operators that follow are useful in performing operations with two vectors, or with a vector and a scalar, where the vector is based on the Vector class. Vector Addition: The + Operator This addition operator adds vector v to vector u according to the formula Here's the code: inline Vector operator+ (Vector u, Vector v) { return Vector(u.x + v.x, u.y + v.y, u.z + v.z); } Vector Subtraction: The - Operator This subtraction operator subtracts vector v from vector u according to the formula Here's the code: inline Vector operator-(Vector u, Vector v) { return Vector(u.x - v.x, u.y - v.y, u.z - v.z); } Vector Cross Product: The Operator This cross product operator takes the vector cross product between vectors u and v, u x v, and returns a vector perpendicular to both u and v according to the formula The resulting vector is perpendicular to the plane that contains vectors u and v. The direction in which this resulting vector points can be determined by the righthand rule. If you place the two vectors, u and v, tail to tail as shown in Figure A-7 and curl your fingers (of your right hand) in the direction from u to v, your thumb will point in the direction of the resulting vector. Figure A-7. Vector Cross Product In this case the resulting vector points out of the page along the z-axis, since the vectors u and v lie in the plane formed by the x- and y-axes. If two vectors are parallel, then their cross product will be zero. This is useful when you need to determine whether or not two vector are indeed parallel. The cross product operation is distributive; however, it is not commutative: Here's the code that takes the cross product of vectors u and v: inline Vector operator^ (Vector u, Vector v) { return Vector( u.y*v.z - u.z*v.y, -u.x*v.z + u.z*v.x, u.x*v.y - u.y*v.x ); } Vector cross products are handy when you need to find normal (perpendicular) vectors. For example, when performing collision detection, you often need to find the vector normal to the face of a polygon. You can construct two vectors in the plane of the polygon using the polygon's vertices and then take the cross product of these two vectors to get normal vector. Vector Dot Product: The * Operator This operator takes the vector dot product between the vectors u and v, according to the formula The dot product represents the projection of the vector u onto the vector v as illustrated in Figure A-8. Figure A-8. Vector Dot Product In this figure, P is the result of the dot product, and it is a scalar. You can also calculate the dot product if you the know the angle between the vectors: Here's the code that takes the dot product of u and v: // Vector dot product inline float operator*(Vector u, Vector v) { return (u.x*v.x + u.y*v.y + u.z*v.z); } Vector dot products are handy when you need to find the magnitude of a vector projected onto another one. Going back to collision detection as an example, you often have to determine the closest distance from a point, which may be a polygon vertex on one body (body 1), to a polygon face on another body (body 2). If you construct a vector from the face under consideration on body 2, using any of its vertices, to the point under consideration from body 1, then you can find the closest distance of that point from the plane of body 2's face by taking the dot product of that point with the normal vector to the plane. (If the normal vector is not of unit length, you'll have to divide the result by the magnitude of the normal vector.) Scalar Multiplication: The * Operator This operator multiplies the vector u by the scalar s on a component-by-component basis. There are two versions of this overloaded operator depending on the order in which the vector and scalar are encountered: inline Vector operator*(float s, Vector u) { return Vector(u.x*s, u.y*s, u.z*s); } inline Vector operator*(Vector u, float s) { return Vector(u.x*s, u.y*s, u.z*s); } Scalar Division: The / Operator This operator divides the vector u by the scalar s on a component-by-component basis: inline Vector operator/(Vector u, float s) { return Vector(u.x/s, u.y/s, u.z/s); } Triple Scalar Product This function takes the triple scalar product of the vectors u, v, and w according to the formula Here, the result, s, is a scalar. The code is as follows: inline float TripleScalarProduct(Vector u, Vector v, Vector w) { return float( (u.x * (v.y*w.z - v.z*w.y)) + (u.y * (-v.x*w.z + v.z*w.x)) + (u.z * (v.x*w.y - v.y*w.x)) ); }

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