Core 4 - Vectors Flashcards
When finding the scalar product of two lines, which part of the vector equation do you use?
Direction vector.
If we know two lines intercept at 90 to each other, what must the scalar product be?
0
If we know two lines are parallel, what can we say about the direction vectors, and the scalar product?
The direction vectors are multiples of each other, the position vectors may be different.
a.b=|a||b|, as cos(0)=1
To work out whether two lines intercept, what do we do to the vector equations?
Equate the x,y and z equations, for both lines, and solve two simultaneously. If the values of lambda and mu work in the third equation, the lines intersect.
What if two of the simultaneous equations solve but the values don’t work in the third?
The lines are skewed, so only appear to intersect when viewed from the third axis. They may also be parallel, so check whether the direction vectors are parallel
How do you form a vector equation of a line between two points?
r=position vector (any point on the line) + (lamda)*(direction vector)
How do you find the direction vector?
Subtract the co-ordinates of one point from the other.
How do you show two lines are skew?
equate two equations, using lambda and mu, solve two equations simultaneously, these will not solve the third. give statement
Must show direction vectors not multiples so not multiples.
When finding the coordinates of a point knowing. Two sides of triangle intersect at right angles, what is the method?
Find equations for the two sides, using a common point e.g. BC and BA. Then find the dot prodictt
For the love of God, do not try to use co_ordinates of points. Must be direction vectors.
If a line meets the xz plane, what can we say about the position vector at this point?
y=0
What do you have to remember when finding the length of a line?
Substitute the value of lambda into the correct equation (FOR THE LINE, NOT THE POINT)
How do you find the angle BAC in a triangle? Which vectors are used?
Both going towards/away from A i.e. AB and AC
What are the two ways to find the perpendicular distance from point to line?
Express point on line in terms of lamda, find vector equation of point-(POL), dot product with vector equation of line
OR
use trig. If you can form a triangle, find the angle between two lines that intersect, and length of one of them
If a point is in two positions on the line and is such that its length is equal to another length so the resulting triangle is isoscolese, how do you find the position of the two points?
Find the length of the line required, (AP^2), without rooting, then find an equation of the line the points lie on in terms of a parameter. Square x y and z and to find (BP^2) and equate to the length (AP^2). Solve the resulting quadratic to find the two lambda solutions, then sub into equation of line to find point.
If finding the coordinates of a point on a trapezium, how do you know which value of mu is used?
Where is the starting point the the equation for the coordinate? If shorter side of trapezium, must be smaller my value, or would be a rhombus.
