Q. 61 Let x1t, x2t, y1t, and y2t聽be d... [FREE SOLUTION]

Chapter 11: Q. 61 (page 862)

Let $x_{1} (t)$ , $x_{2} (t)$ , $y_{1} (t)$ , and $y_{2} (t)$ be differentiable scalar functions. Prove that the dot product of the vector-valued functions role="math" localid="1649579098744" $r_{1} (t) = <x_{1} (t), y_{1} (t)>$ and role="math" localid="1649579122624" $r_{2} (t) = <x_{2} (t), y_{2} (t)>$ is a differentiable scalar function.

Short Answer

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The dot product of the vector-valued functions $r_{1} (t) = <x_{1} (t), y_{1} (t)>$ and $r_{2} (t) = <x_{2} (t), y_{2} (t)>$ is a differentiable scalar function.

Step by step solution

Step 1. Given information

We have to prove that the dot product of the vector-valued functions $r_{1} (t) = <x_{1} (t), y_{1} (t)>$ and $r_{2} (t) = <x_{2} (t), y_{2} (t)>$ is a differentiable scalar function.

Step 2. Proof of the question.

$r_{1} (t) 路 r_{2} (t) = <x_{1} (t), y_{1} (t)> 路 <x_{2} (t), y_{2} (t)> = (x_{1} (t) i + y_{1} (t) j) 路 (x_{2} (t) i + y_{2} (t) j) = x_{1} (t) x_{2} (t) (i 路 i) + x_{1} (t) y_{2} (t) (i 路 j) + y_{1} (t) x_{2} (t) (j 路 i) + y_{1} (t) y_{2} (t) (j 路 j) = x_{1} (t) x_{2} (t) + y_{1} (t) y_{2} (t)$

Therefore,

$r_{1} (t) 路 r_{2} (t) = x_{1} (t) x_{2} (t) + y_{1} (t) y_{2} (t)$ , is a scalar function.

The product and sum of differentiable functions are differentiable.

Since $x_{1} (t)$ and $x_{2} (t)$ are differentiable $鈬� x_{1} (t) x_{2} (t)$ is differentiable.

$y_{1} (t), y_{2} (t)$ are differentiable $y_{1} (t) y_{2} (t)$ is differentiable.

Finally,

$x_{1} (t) x_{2} (t) + y_{1} (t) y_{2} (t)$ is differentiable.

Hence $r_{1} (t) 路 r_{2} (t)$ is a differentiable scalar function.

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91影视

Let x1t, x2t, y1t, and y2tbe differentiable scalar functions. Prove that the dot product of the vector-valued functions role="math" localid="1649579098744" r1t=x1t,y1tand role="math" localid="1649579122624" r2t=x2t,y2tis a differentiable scalar function.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Proof of the question.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Logic and Functions

Mechanics Maths

Applied Mathematics

Pure Maths

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.