Test 1 Flashcards
Two vectors are parallel in the same direction if a^=
c*b^ and c>0
Unit vector =
direction/magnitude
Find a vector length 2 in the same direction as 3i^-4j^
2(unit vector for 3i^-4j^)
How to take dot product
multiply corresponding components and then sum all together
Dot product results in a…
scalar quantity
The dot product helps find…
the angle between 2 vectors
What equation helps find the angle between 2 vectors?
cos(theta) = (a^ dot b^)/(mag(a)*mag(b))
a^ and b^ are perpendicular if
dot product is 0
How to prove <2,4> and <6,-3> are orthogonal
Take dot product. It would equal 0.
Statics use for cross product
M = r^ x F^
det |ab/cd| =
ad-bc
How to take cross product
i^ j^ k^ / a1^ a2^ a3^ / b1^ b2^ b3^ |
cofactor expansion
i^det-j^det+k^det
sign for i,j,k
(-1)^row+column
a^ x b^ is ______________ to the plane formed by vectors a^ and b^
perpendicular
if a^ x b^ = 0, then
a^ is parallel to b^
length of cross product can be calculated by
finding the area of the parallelogram formed by a^ and b^, which is || a^ x b^ ||
1/2 || a^ x b^ || =
area of triangle formed by a^ and b^
What do you need to plot a line in 3D?
Point (x1, y1, z1) and parallel vector (slope)
What parametric equations are used to plot a line?
x = x1 + a1t y = y1 + a2t z = z1 + a3t Point (x1, y1, z1) Parallel vector
How do you plot a line given two points?
P2-P1 gives parallel vector. Use P1 as point.
What do you need to plot a plane in 3D?
1) point from shared tail
2) perpendicular vector found from cross product
Given three points. Steps to find equation of plane:
1) Subtract points to find vectors
2) Take cross product of vectors (gives perpendicular vector) (n^=<i>)
3) plugs values of equation using n vector and point from shared tail
n1(x-x1)+n2(x-x2)+n3(x-x3)=0</i>
plane point normal form
-18(x-3)+0(y-2)+24(z-1)=0
plane standard form
-18x+24z+30=0
Two planes are parallel if their normal vectors are…
parallel
two planes are perpendicular if their normal vectors are
perpendicular
Vectors associated with lines are
parallel
Vectors associated with planes are
perpendicular
Find the normal vector from the equation of the plane.
2x-3y-z-5=0
n^ = <2, -3, -1>
How can you tell if planes are parallel or perpendicular?
If the normal vectors are multiples, then they are parallel. If dot product is 0, they are perpendicular.
Which equations should you set equal to one?
Ellipsoid, cone, and paraboloid
x^2+y^2=16 is what kind of equation about which axis?
Cylinder about the z-axis
x=-1 in 3D is what kind of equation?
Plane
What does square rooting an equation do?
Make it either a positive or negative half
