Vectors Flashcards
what is a scalar
a quantity with a magnitude only
what is a vector
a quantity with magnitude and direction
what is a zero vector
any vector with all components zero
how do we represent magnitude for vector u?
|u|
how to calculate the magnitude of a vector
the square root of all the components squared
what is a unit vector?
any vector with a magnitude of one
what is the 3-dimensional version of the distance formula?
d = square root of (x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2
what are equal vectors?
vectors with the same magnitude and direction
all components are equal
what position do vectors need to be in to add or subtract them?
nose to tail
what is the negative of a vector?
the vector multiplied by 1
what is OA (with arrow above)
a position vector of point A relative to the origin O, and is written as a
what can ai + bj + ck be written as?
(a b c)
the points A, B and C in 3D space are collinear if…
AB is parallel to BC, with B a common point
non-zero vectors are parallel if…
they are scalar multiples of the same vector
steps to find the coordinate of a point which divides a line segment in a given ratio
- make a sketch
- equate the ratio of the two lines with the given ratio
- cross multiply and change line segments into position vectors
- rearrange
- write as a coordinate
scalar product formula
a.b = |a| |b| cos theta
what do you need to remember when calculating scalar product
the vectors a and b need to both either be pointing towards or away from the angle
if one is pointing away while the other points towards then a.b = -|a| |b| cos theta
the component form of the scalar product
a.b = a1b1 + a2b2 + a3b3
how to calculate the angle between vectors
cos theta = a.b / |a||b|
or
cos theta = a1b1 + a2b2 + a3b3 / |a||b|
what is the scalar product when the vectors are perpendicular
0
properties of the scalar product
a. b = b.a
a. (b+c) = a.b + a.c
a. a = |a|^2
