Final answer:
a) -3p - 4q = -5i - 21j, b) |p + q| = sqrt(37), c) The angle between p and q = arccos(1 / sqrt(130)), d) The scalars h and o are 4 and -1.
Step-by-step explanation:
To find the results for each part of the question, we can use the given position vectors.
a) -3p - 4q = (-3)(-i+3j) - (4)(2i+3j) = 3i - 9j - 8i - 12j = -5i - 21j
b) |p + q| = |(-i+3j) + (2i+3j)| = |i + 6j| = sqrt((1)^2 + (6)^2) = sqrt(37)
c) To find the angle between two vectors, we can use the dot product formula.
The formula is cos(theta) = (p · q) / (|p||q|). Given p = -i+3j and q = 2i+3j, the dot product (p · q) = (-1)(2) + (3)(3) = 1.
The magnitudes |p| and |q| can be found using the formula sqrt((i)^2 + (j)^2).
So |p| = sqrt((-1)^2 + (3)^2) = sqrt(10) and |q| = sqrt((2)^2 + (3)^2) = sqrt(13).
Therefore, the angle theta = arccos(1 / (sqrt(10) * sqrt(13))) = arccos(1 / sqrt(130))
d) To find scalars h and o such that hp + kq = 5i + 7j, we can set the coefficients of i and j equal to the corresponding coefficients on both sides of the equation.
This gives us -h + 2o = 5 and 3h + 3o = 7. Solving these equations simultaneously, we get h = 4 and o = -1.
Therefore, the scalars h and o are 4 and -1.