vectors 2 space
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directed line segment
how is bearing different from trig angles?
- bearing north = 0, goes clockwise
- trig east = 0, goes counterclockwise
two ways to subtract a - b
- either tail to tail
- or a + (-b)
what's an easy way to add vectors ab + bc?
like matrices, cross out b, answer = ac
if v is a vector, what is kv?
- also a vector
- if k > 0, same direction
- if k = 0, 0 vector
- if k < 0, opposite direction
- |kv| = |k||v|
vectors that are scalar multiples of one another
how do you find a linear combination when given two vectors?
- as long as a and b are not collinear
- if given a and b to find c
- c = ka + mb
- write in cartesian form then use system of equations
how do you convert from two points to vector?
head - tail, x and y coordinates
how to find magnitude of vector a?
|a| = sqrt(a12 + a22 )
how do you determine if two vectors are collinear?
if they can become one another by dividing/multiplying by scalar
how to find a vector with magnitude 1 when given vector a?
- |ka| = |k||a|
- |k||a| = 1
- |k|| sqroot formula| = 1
- k = 1/sqroot formula
- then multiply k with a
word problems two solutions
- geometric, use cosine law
- cartesian, use pythagorean theorem to find angle and convert to cartesian vectors
if using trig what must you always do?
- convert bearing angle to trig angle
- trig angle = 450 - bearing angle
how to convert to component form? for vector a
= [magnitude*cos degree, magnitude *sin degree]
force that counterbalances resultant, equal in mag. opposite in direction
two methods for forces
- use sine law with magnitude resultant/sin angle when asked to find the magnitudes of two ropes
- pay attention to F and C patterns
- cartesian method: use magnitudes and variables to represent the t1 and t2 then use substitution method and system of equations
two formulas for dot product
- a * b = |a||b|costheta
- a * b = a1b1 + a2b2
what does the dot product give and reveal?
- dot product = scalar
- if dot product greater than 0, = acute angle
- dot product smaller than 0, obtuse
- if = 0, 90 degree angle
to find the angle using dot product, what formula?
a1b1 + a2b2 = |a||b|costheta
properties o dot product
- a * a = |a|2
- a * 0(vector) = 0(scalar)
- |a+b|2 = |a|2 + 2a * b + |b|2
- |a+b|2 = (a+b)*(a+b)
- a * u = |a| if u is unit vector w/ same direction as a
- be careful when simplifying write out with x and y
if in a triangle there are the same dot product for angles what does it mean?
the angles are the same!
what is the formula for projects? (2)
- if a on b
- O = (a*b)/(b*b)
- O = (a*b)/(|b|2 )
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