Q. 17

Question

What is the dot product of the vector functions $r_{1} (t) = (x_{1} (t), y_{1} (t)) and r_{2} (t) = (x_{2} (t), y_{2} (t)) ?$

Step-by-Step Solution

Verified

Answer

The dot product of the given two vector functions is $x_{1} (t) x_{2} (t) + y_{1} (t) y_{2} (t) .$

1Step 1. Given Information.

The given vector functions are $r_{1} (t) = (x_{1} (t), y_{1} (t)) and r_{2} (t) = (x_{2} (t), y_{2} (t)) .$

2Step 2. Find the dot product.

Let's find the dot product,

$r_{1} (t) \cdot r_{2} (t) = (x_{1} (t), y_{1} (t)) \cdot (x_{2} (t), y_{2} (t)) = (x_{1} (t) i + y_{1} (t) j) \cdot (x_{2} (t) i + y_{2} (t) j) = x_{1} (t) x_{2} (t) (i \cdot i) + x_{1} (t) y_{2} (t) (i \cdot j) + y_{1} (t) x_{2} (t) (j \cdot i) + y_{1} (t) y_{2} (t) (j \cdot j) [Since i \cdot i = 1, i \cdot j = 0, j \cdot i = 0, j \cdot j = 1] = x_{1} (t) x_{2} (t) + y_{1} (t) y_{2} (t)$

Thus, the product of two vector functions is a scalar function.

Q. 16

Q. 18

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Q. 18

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Q. 19

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