Find two unit vectors in 2-space that make an angle of 45◦ with 4i+3j.

First, lets transform the given vector into an unit vector (dividing by its module)

UnitVec = 4/5 i + 3/5 j

Then lets change this vector into a polar form

UnitVec = 1. with angle of 36.869 degrees taking as a reference the i vector

Then, the probem tells us that the vectors u and v make an angle of 45 degrees with UnitVec, so lets add+-45 to the vector in polar form

U = 1*[cos(36.869 +45)i + sin(36.869 +45)j] = 0.1414 i + 0.9899 j

V = 1*[cos(36.869 -45)i + sin(36.869 -45)j] = 0.9899 i - 0.1414 j

Intersecting secant-tangent theorem states that if a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the measure of the tangent segment is equal to the product of the measures of the secant segment and its external secant segment.so to break it down: 1. pq acts as the tangent and ps as the secant with an intersection at p.sr = 21rp = 3x + 3pq = 4x + 4following to equation: tangent squared = the two segments multiplied by each otherso (4x + 4) ^ 2 = (3x + 3) • (21) simplifies to 16x^2 - 31x - 47 = 0 or (16x - 47)(x + 1) x = -1 or x = 47/162. pq acts as the tangent and ps as the secant with an intersection at p (again! )sr = 16rp = xpq = 15(15) ^ 2 = 16 • x225 = 16xx = 225/163. the interesting chords are proportional to each other so a ratio is possible to set up: ap dp 10 3 + x = = —— = pc pb 8 xcross multiple to get 10x = (3 + x)(8)x = 12

0=6x+2y+10 is equivalent to the simplified form: y= -3x - 5• parallel lines always have the same slope, different y intercept knowing this, we can say the slope, or m of the line parallel to y= -3x - 5 is -3.m= -3 hope this could !