Math 126 Final Study Guide Grace Qiu 1 Basic Vectors Magnitude of a vector Dot product MIDTERM 1 MATERIAL Vectors have two components a direction and a magnitude If r r1 r2 r3 Magnitude r a b a1b1 a2b2 a3b3 Result a scalar When this equals 0 the two vectors are orthogonal to each other Angle is pi 2 When a b a b parallel Angle is 0 or pi a b a b cos where is the acute angle between a and b Cross product of two vectors 1 2 3 1 2 3 a b a2b3 a3b2 i a1b3 a3b1 j a1b2 a2b1 k Result a vector When this equals the zero vector the two vectors are parallel to each other a b a b sin area of parallelogram Angle between two vectors a b a b cos where is the acute angle between a and b a b a b sin where is between a and b and 0 Vector given by line segment AB A x1 y1 and B x2 y2 Two vectors are also parallel to each other if they are scalar multiples of each other a x2 x1 y2 y1 Components projab Parallel to a but points in opposite direction compab projab Parallel to a but points in same direction Math 126 Final Study Guide Grace Qiu 2 Basic equations Describing a Line You need a point and a direction vector Line equations are NOT UNIQUE scalar multiples exist Parametric equation x 1 4t y 2 5t z 3 6t Vector r t 1 4t 2 5t 3 6t Symmetric Describing a Plane You need a point and a normal vector to the plane Point P xo yo zo and vector a b c a x xo b y yo c z zo 0 Plane equations ARE UNIQUE Solve each parametric equation for t and set equal 3 Finding equations Equation of a plane passing through three given points Create two lines from the three given points Find the cross product of these two lines this will give you a vector orthogonal to these The result is the normal vector to the plane You can then use this and any of the given points to construct an equation of a plane Equation of a plane passing through a point parallel to a given plane The normal vector of the given plane is the normal vector of the unknown plane Construct equation using the point and this vector Equation of a plane containing a line and a given point Make one vector the coefficients of the line equation To get the second vector set the variable in the line equation to 0 This will give you a point Use this point and the given point to make an equation for a vector Do the cross product of this new vector and the first vector you made This will give you the normal vector to the plane Use the given point and this normal vector to construct an equation Math 126 Final Study Guide Grace Qiu 4 Distance Distance between two points P1 P2 Do three points make a triangle or a line 2 1 2 1 2 1 Use the three points and find the longest distance A to B A to C B to C If smaller distances add up to the longest distance they are in a line If not they are in a triangle Equilateral Isosceles Look at the distances Right triangle Only if this is true a2 b2 c2 Distance between a point and a plane Distance from a point on a line to a plane v t x t y 1 z 5 from plane y 5 Process Make an arbitrary point P x y z 5 Distance from P to the plane y 5 only thing that matters is P s y coordinate Distance between two parallel planes Midpoint of a line Center and radius of a sphere given by equation x h 2 y k 2 z l 2 r2 Center is at h k l with radius r 1 5 Distance from P to the line v t Since x t in v t that means the value of point P on line v t is x Math 126 Final Study Guide Grace Qiu 5 Intersection Point of intersection of two lines Make the variables different such as one with s and one as t Set parametric equations or other equal to each other accordingly Use substitution to solve for your two unknowns Plus the s value back into the equation with s s or t back into the equation with t s to find the specific point Check they should come out to be the same Line of intersection of two planes n1 n2 v The equations of the two planes give you normal vectors orthogonal to each plane Finding the cross product gives you a vector orthogonal to the normal vector which means it is parallel to the planes A line lying on both planes will be parallel to both Let a variable 0 in the plane equations and use substitution to solve for the remaining two variables This is using the assumption that at some point the planes will intersect the xy plane the yz plane or the xz plane Angle between two intersection planes cos 1 If comes out to be negative you chose normal vectors that created an angle bigger than pi 1 If this is the case simply do cos 1 or Angle between two intersecting lines Find the vectors for both a b a b cos where is the acute angle between a and b Find the equation of a plane containing a line of intersection of two other planes and a given point Get the two normal vectors of the two planes Do their cross product to get the vector for the line of intersection Find a point on the line of intersection by setting a variable to 0 in the plane equations and then solve this system of equations This results in a point Using this found point and the given point create a vector Do the cross product of this new vector and the vector of the line of intersection This gives the normal vector to the unknown plane Use this normal vector and either the given point or found point to construct an equation Math 126 Final Study Guide Grace Qiu 6 Miscellaneous Questions Whether or not a point is a part of a given line plane Put point coordinates into line plane equation If there is ONE variable that satisfies all equations of line equation of plane it exists on the line plane If not it does not Whether or not a line is part of a given plane Put line equations into the respective x y and z variables If there is ONE variable that satisfies this new plane equation then yes it is part of the given plane Whether or not two planes intersect or are parallel If two planes normal vectors are parallel scalar multiples of each other using cross product etc then planes are parallel If not parallel then find the …
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