Find the angle between \( u \) and \( v \) in radians. \( u=-4 j \) and \( v=6 i-7 k \) \( 1.68 \) \( 0.00 \) \( 0.11 \) \( 1.57 \)
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To find the angle between \( u \) and \( v \) in radians, we can use the dot product formula:
\[ \cos(\theta) = \frac{u \cdot v}{|u||v|} \]
First, let's find the dot product \( u \cdot v \):
\[ u \cdot v = (-4j) \cdot (6i-7k) = 0+0+(-4)(-7) = 28 \]
Next, let's find the magnitudes of \( u \) and \( v \):
\[ |u| = |-4j| = \sqrt{0+(-4)^2+0} = 4 \]
\[ |v| = |6i-7k| = \sqrt{(6)^2+0+(-7)^2} = \sqrt{36+0+49} = \sqrt{85} \]
Now we can substitute these values into the dot product formula:
\[ \cos(\theta) = \frac{u \cdot v}{|u||v|} = \frac{28}{4\sqrt{85}} = \frac{7}{\sqrt{85}} \]
To find the angle \( \theta \), we can take the inverse cosine of this value:
\[ \theta = \cos^{-1}\left(\frac{7}{\sqrt{85}}\right) \approx 1.68 \]
Therefore, the angle between \( u \) and \( v \) in radians is approximately \( \boxed{1.68} \).
To find the angle between vectors \( u \) and \( v \), we can use the dot product formula:
\[ \cos(\theta) = \frac{{u \cdot v}}{{\| u \| \cdot \| v \|}} \]
First, let's calculate the dot product of vectors \( u \) and \( v \):
\[ u \cdot v = (-4)(0) + (0)(6) + (-4)(-7) = 0 + 0 + 28 = 28 \]
Next, let's calculate the magnitudes of vectors \( u \) and \( v \):
\[ \| u \| = \sqrt{(-4)^2 + 0^2 + 0^2} = \sqrt{16} = 4 \]
\[ \| v \| = \sqrt{6^2 + 0^2 + (-7)^2} = \sqrt{36 + 49} = \sqrt{85} \]
Now, let's substitute those values into the formula to find \( \cos(\theta) \):
\[ \cos(\theta) = \frac{28}{4 \cdot \sqrt{85}} = \frac{7}{\sqrt{85}} \approx 0.805 \]
To find \( \theta \), we can take the inverse cosine of \( \cos(\theta) \):
\[ \theta = \cos^{-1}(0.805) \approx 0.6447 \]
The angle between \( u \) and \( v \) in radians is approximately \( 0.6447 \).