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Unformatted text preview: Vectors • Vectors represent a quantity a direction For starters, we can think in two dimensions, ~ A = A x ˆ i + A y ˆ j We have here ˆ i and ˆ j as unit vectors in the x and y directions respectively. By unit vector, we mean they have length=1. We can add vectors algebraically or numerically, for example if ~ B = B x ˆ i + B y ˆ j , then ~ A + ~ B = ( A x + B x ) ˆ i + ( A y + B y ) ˆ j If we have three dimensions, we need ˆ k as a unit vector along z in a Cartesian system. Patrick K. Schelling Introduction to Theoretical Methods Magnitude of a vector This is just the length of a vector. For example in two dimensions, we use the Pythagorean theorem, A =  ~ A  = q A 2 x + A 2 y In three dimensions, the vector ~ A is ~ A = A x ˆ i + A y ˆ j + A z ˆ k and the generalization to find the magnitude is obvious, A =  ~ A  = q A 2 x + A 2 y + A 2 z Patrick K. Schelling Introduction to Theoretical Methods Scalar product or dot product We define the dot product, or scalar product as, ~ A · ~ B =  ~ A  ~ B  cos θ where cos θ is the angle between the two vectors in the plane they form. As components, we write the dot product, for example in three dimensions, ~ A · ~ B = A x B x + A y B y + A z B z Patrick K. Schelling Introduction to Theoretical Methods Vector or cross product The cross product or vector product has magnitude,  ~ A × ~ B  =  ~ A  ~ B  sin θ However, ~ A × ~ B actually is itself a vector! We define the direction of ~ C = ~ A × ~ B perpendicular to the plane made by ~ A and ~ B . This means: I ~ A × ~ B = 0 if ~ A and ~ B are parallel/antiparallel ( θ = 0 or θ = π ) Patrick K. Schelling Introduction to Theoretical Methods Vector or cross product The cross product or vector product has magnitude,  ~ A × ~ B  =  ~ A  ~ B  sin θ However, ~ A × ~ B actually is itself a vector!...
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 Spring '03
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 Dot Product, theoretical methods, Patrick K. Schelling

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