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# so far - So far in our course Important theoretical...

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So far in our course Important theoretical concepts The vector v that has originates at the point ( a 1 , b 1 ) and terminates at the point ( a 2 , b 2 ) has components < a 2 - a 1 , b 2 - b 1 > . The magnitude of the vector v = < a, b > equals: || v || = p a 2 + b 2 The sum of the vectors u = < u 1 , u 2 > and v = < v 1 , v 2 > has compo- nents: u + v = < u 1 + v 1 , u 2 + v 2 > The scalar multiple λ v of the vector v = < v 1 , v 2 > has components: λ v = < λv 1 , λv 2 > The vectors u and v are parallel if and only if there exists real number λ such that u = λ v or v = λ u . A vector u is unit if || u || = 1. If v is a non-zero vector, then v / || v || is a unit vector. All the above concepts generalize trivially to 3-dimensional vectors. The dot product of two 3-dimensional vectors u = < u 1 , u 2 , u 3 > and v = < v 1 , v 2 , v 3 > is defined by the formula: u · v = u 1 v 1 + u 2 v 2 + u 3 v 3 The dot product is linear with respect to each of the two variables and commutative. Also, notice that uu = || u || 2 for any vector u . Two vectors are orthogonal to each other if and only if their dot prod- uct is 0. 1

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The angle θ between two vectors u and v is defined by the formula: cos θ = u · v || u |||| v || The projection of a vector u onto a non-zero vector v is given by the formula: proj v ( u ) = u · v v · v v The component of a vector u along a non-zero vector v is given by the expression: u · v || v || For any two vectors u and v we have the following inequality: | u · v | ≤ || u |||| v || The equality holds true if and only if u and v are parallel. The cross product of two three dimensional vectors u = < u 1 , u 2 , u 3 > and v = < v 1 , v 2 , v 3 > is defined by the formula: u × v = i j k u 1 u 2 u 3 v 1 v 2 v 3 The vector u × v is perpendicular to both u and v . Also, the vectors u , v , u × v form a right-handed system. The cross product is linear with respect to each of the two variables but it is not commutative. In fact, it satisfies the relation: u × v = - ( v × u ) Given 3 vectors u = < u 1 , u 2 , u 3 > , v = < v 1 , v 2 , v 3 > and w = < w 1 , w 2 , w 3 > , their mixed product is defined by the formula: ( u × v ) · w 2
It is actually a number and it equals the determinant: u × v = u 1 u 2 u 3 v 1 v 2 v 3 w 1 w 2 w 3 The absolute value of this determinant equals the volume of the par- allelepiped formed by the three vectors. The magnitude of the cross product of the vectors u and v equals the area of the parallelogram formed by these two vectors. In particular: || u × v || = || u |||| v || sin θ where θ is the angle between the two vectors. The equation of the plane that goes through the point ( x o , y o , z o ) and is perpendicular to the non-zero vector < a, b, c > is given by: < a, b, c. > · < x - x o , y - y o , z - z o > = 0 or equivalently: a ( x - x o ) + b ( y - y o ) + c ( z - z o ) = 0 If we wish to find the equation of a plane that goes through three points, then all we need to do is find two linearly independent vectors that go through these three points and take their cross product to obtain a vector which is orthogonal to the plane. Then we are back in the case of a plane that goes through a given point (in fact we have three points to choose from) and is perpendicular to a given plane.

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so far - So far in our course Important theoretical...

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