Geometric Objects. 2001. 7. 6 Computer Graphics Lab. Sun-Jeong Kim. Points. Single Coordinate Position Set the bit value(color code) corresponding to a specified screen position within the frame buffer. y. setPixel ( x , y ). x. Lines. Intermediate Positions between Two Endpoints

ByVectors. Recall that “vectors” are arrows that represent a vector quantity (magnitude and direction). The length of the arrow represents the magnitude of a measurement The direction of the arrow within a coordinate system represents the direction of the vector.

ByVector Product (Cross-product). There are two different methods for determining the vector product between any two vectors: The Determinant method and the Cyclic method. Determinant Method. Cyclic Method.

ByWebsite. Mechanics. Kinematics and Forces. Important things to remember. 1) Units – Every numerical quantity must have units associated with it!!. Significant Figures – Important to show how precisely you know a measured quantity or a quantity calculated from measurements.

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Vector Product (Cross-product). There are two different methods for determining the vector product between any two vectors: The Determinant method and the Cyclic method. Determinant Method. Cyclic Method.

The Vector Cross Product. Definition. The cross product of two 3-d vectors becomes a 3-d vector itself. The cross product is:. Example. Method 1. An easier method is to use the formula listed in the matrices part of the formula sheet. Method 2.

The Vector or Cross Product. Lecture V1.3 Example 5 Moodle. Definition of the Cross Product. commutative law does NOT hold. Distributive Law. Cross Product and Vector Components. 0. Cross Product and Vector Components. Associative Law. Does NOT, in general, hold for the cross product.

Vector Multiplication: The Cross Product. When two vectors are “multiplied” to form a 3 rd vector, the new vector is called the cross product of the original vectors. Symbolically, you write. The magnitude of the cross product is given by.

Derivation of the Vector Dot Product and the Vector Cross Product. Derivation of the Vector Dot Product. u·v =∑ i u i v i = ∑ i u i e i ∑ i v j e j. (u 1 e 1 + u 2 e 2 + u 3 e 3 ) (v 1 e 1 + v 2 e 2 + v 3 e 3 ). Kronecker Delta e i ·e j = δ ij = 1 when i = j

Derivation of the Vector Dot Product and the Vector Cross Product. Derivation of the Vector Dot Product . u·v =∑ i u i v i = ∑ i u i e i ∑ i v j e j. (u 1 e 1 + u 2 e 2 + u 3 e 3 ) (v 1 e 1 + v 2 e 2 + v 3 e 3 ). Kronecker Delta e i ·e j = δ ij = 1 when i = j

Vector Product. Results in a vector. Dot product (Scalar product). Results in a scalar a · b = a x b x +a y b y +a z b z. Scalar. Vector Product. Results in a vector. =. Properties…. a x b = - b x a a x a = 0 a x b = 0 if a and b are parallel.

y. q. j. i. x. i. k. z. j. k. Cross Product . The cross product of two vectors says something about how perpendicular they are. Magnitude: is smaller angle between the vectors Cross product of any parallel vectors = zero Cross product is maximum for perpendicular vectors

Cross Product. Before discussing the second way to “multiply” vectors, we need to talk about matrices… If , then the determinant of A is | A | = ad – bc To find the determinant of a 3 3 matrix, we will work along the top row. Ex.

Cross Product. Ali Tamaki Ben Waters Linear Systems Spring 2006. Definition. The cross product is defined as the vector perpendicular to two given vectors A and B separated by an angle and is shown by: ||A x B|| = ||A|| ||B|| sin