Given that \(q = 9i + 6j\) and \(r = 4i - 6j\), which of the following statements is true?
Answer Details
To determine which of the statements is true, we can use the properties of vectors.
We start by finding the magnitude of vector r using the formula:
|magnitude of r| = sqrt(x^2 + y^2)
where x and y are the coefficients of the i and j terms respectively.
|magnitude of r| = sqrt(4^2 + (-6)^2) = sqrt(52)
So the statement "The magnitude of r is 52 units" is true.
Next, we can find the dot product of vectors q and r. If the dot product is equal to zero, then the vectors are perpendicular. If the dot product is non-zero, then the vectors are not perpendicular.
q . r = (9 * 4) + (6 * -6) = 36 - 36 = 0
Since the dot product of q and r is zero, the statement "r and q are perpendicular" is true.
Therefore, the correct answer is:
- r and q are perpendicular.