Find correct to the nearest degree,5 the angle between p = 12i - 5j and q = 4i +3j
Answer Details
To find the angle between two vectors, we can use the dot product formula:
p \(\cdot\) q = \|p\| \|q\| cos \(\theta\)
where p \(\cdot\) q is the dot product of vectors p and q, \|p\| and \|q\| are the magnitudes of vectors p and q respectively, and \(\theta\) is the angle between the two vectors.
First, let's calculate the magnitudes of vectors p and q:
\|p\| = \(\sqrt{(12)^2 + (-5)^2}\) = \(\sqrt{169}\) = 13
\|q\| = \(\sqrt{(4)^2 + (3)^2}\) = \(\sqrt{25}\) = 5
Next, let's calculate the dot product of vectors p and q:
p \(\cdot\) q = (12)(4) + (-5)(3) = 48 - 15 = 33
Substituting the values we obtained into the formula for the dot product, we get:
33 = (13)(5) cos \(\theta\)
Solving for cos \(\theta\), we get:
cos \(\theta\) = \(\frac{33}{65}\)
Using a calculator, we can find that the inverse cosine of \(\frac{33}{65}\) is approximately 59.08\(^o\).
Therefore, the angle between vectors p and q is approximately 59 degrees when rounded to the nearest degree.
Answer: 59\(^o\).