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Understanding Displacement Vectors and Vector Operations in Calculus III

Explore displacement vectors, dot and cross products, vector properties, and applications in Calculus III Chapter 13, with detailed explanations and examples.

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Understanding Displacement Vectors and Vector Operations in Calculus III

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  1. Calculus III Chapter 13 Br. Joel Baumeyer Christian Brothers University

  2. Displacement Vectors • A displacement vector from one point to another is an arrow with its tail at the first point and its tip at the second. • The magnitude or length of the displace- ment is the distance between the points, and is represented by the length of the arrow. • The direction of the displacement vector is the direction of the arrow. • Displacement vectors which point in the same direction and have the same magnitude are considered to be the same, even if they do not coincide.

  3. Vectors and Ordered n-tuples • The standard basis displacement vectors: I, j and k are represented by the ordered triples [1,0,0], [0,1,0] and [0,0,1] respectively. 0 = [0,0,0]. If v = , w = , u = then the following are provable: (a,b - scalars) • u + v = v + u (a +b)v = av + bv • (u + v ) + w = u + (v + w )a(v + w) = av + aw • a(bv) = (ab)v v + 0 = v 1v = v • v + w = + = • av = a = • Norm of a vector: ||v|| =

  4. The Dot Product & Properties • Definition:v×w = ||v|| ||w|| cos q, q the angle between v and w. • Thus q = • If then, • v×(lw)= l(v × w) = (lv)×w; v×w = w×v • (v + w)×u = v×u + w×u; v × v = ||v|| ||v|| • A unit vector u : A vector whose norm (magnitude) is 1. u = v/||v||

  5. Applications of the Dot Product • Work is an example of the dot product: W = F •d • Given two vectors F and d, with d not 0 , then • is parallel to d and is perpendicular to d. • Then : and • The equation of the plane with normal vector n = ai +bj +ck and • containing the point is • Or ax + by + cz = d, if

  6. Cross Product & Properties • Definition: v´w = ||v|| ||w|| sinqn, n a unit vector perpendicular to the plane formed by v and w. • i´j = k,, j´k = i, k´i = j, j´i = -k, k´j = -i, i´k = -j • i´i = 0, j´j = 0, k´k = 0 • v´(lw)= l(v´ w) = (lv)´w • v´w = -w´v; u´(v + w) = u´v+ u´w • .

  7. Applications of the Cross Product The cross product v´w is a vector that has • magnitude equal to the area of the parallelo-gram determined by v and w. • direction perpendicular to plane of v and w and determined by the right-hand rule. • The volume of a parallelepiped (box) formed by the three vectors u, v and w is given by |u×(v´w)| = = |v×(u´w)| = |w×(v´u)|

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