"Prove that if r is a vector-valued function..." solved and explained

TokNeekCepTdh

Answered question

2021-11-26

Prove that if r is a vector-valued function that is continuous at c, then

| | r | |

is continuous at c.

Elizabeth Witte

Beginner2021-11-27Added 24 answers

Step 1
Consider the vector

\vec{r} (a) = p (a) i + q (a) j + s (a) k

If r is continuous at

a = c

. So,

lim_{a \to c} \vec{r} (t) = \vec{r} (c)

Then p(a), q(a) and s(a) are defined at

a = c

.
Step 2
The magnitude of the vector at

a = c

is,

| \vec{r} | = \sqrt{{(p (a))}^{2} + {(q (a))}^{2} + {(s (a))}^{2}}

lim_{a \to c} | \vec{r} | = \sqrt{{(p (c))}^{2} + {(q (c))}^{2} + {(s (c))}^{2}}

= | \vec{r} (c) |

Hence the vector

| \vec{r} |

is continuous at

a = c

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