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Chapter 6: Problem 18

Find a number \(t\) such that the vectors \((2 \cos t, 4)\) and (10,3) are perpendicular.

Short Answer

Expert verified
The value of \(t\) is approximately \(2.2143\) radians for the given vectors to be perpendicular.

Step by step solution

01

Calculate the dot product of the vectors

To find the value of \(t\) such that the vectors \(A = (2 \cos t, 4)\) and \(B = (10, 3)\) are perpendicular, we need to compute their dot product. The dot product of two vectors (\(A\) and \(B\)) in 2D is given by the formula: \[A \cdot B = a_x b_x + a_y b_y\] Applying this formula to our vectors, we have: \(A \cdot B = (2 \cos t) (10) + (4) (3)\)
02

Set the dot product equal to zero and solve for t

Since the vectors are perpendicular, their dot product is zero. So, we set the dot product equal to zero and solve for \(t\). \((2 \cos t) (10) + (4) (3) = 0\) Now we can simplify the equation: \(20 \cos t + 12 = 0\) Now we isolate \(\cos t\): \(\cos t = -\frac{12}{20} = -\frac{3}{5}\)
03

Find the value of t using the inverse cosine function

Now that we have the cosine value, we can find the angle \(t\) using the inverse cosine (or arccos) function: \(t = \arccos{-\frac{3}{5}}\) To get the final value of \(t\), we can either use a calculator to find the arccos value or provide an approximate decimal value: \(t \approx 2.2143\, \text{radians}\) So the value of \(t\) is approximately \(2.2143\) radians for the given vectors to be perpendicular.

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