"Let r(t) and u(t) be vector-valued functions whose..." solved and explained

yapafw

Answered question

2021-11-28

Let

r (t)

and

u (t)

be vector-valued functions whose limits exist as t c. Prove that

lim_{t \to c} [r (t) \times u (t)] = lim_{t \to c} \times lim_{t \to c} u (t)

William Yazzie

Beginner2021-11-29Added 20 answers

Step 1
Given

r (t)

and

u (t)

be vector valued function whose lint exist as

t \to c

To prove

lim_{t \to c} [r (t) \times u (t)] = lim_{t \to c} r (t) \times lim_{t \to c} u (t)

Step 2
Dot product
If

a = a_{1} i + a_{2} j + a_{3} k

and

b = b_{1} i + b_{2} j + b_{3} k

a \times b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}

Step 3
Let

r (t) = a (t) i + b (t) j + c (t) k

u (t) = m (t) i + n (t) j + p (t) k

On taking L.H.S

lim_{t \to c} [r (t) \times u (t)] = lim_{t \to c} (a (t) i + b (t) j + c (t) k) \times (m (t) i + n (t) j + p (t) k)

= lim_{t \to c} (a (t) m (t) + b (t) n (t) + c (t) p (t))

= lim_{t \to c} (a (t) m (t)) + lim_{t \to c} (b (t) n (t)) + lim_{t \to c} (c (t) p (t))

Since limit exist as t tends to c.

= a (c) m (c) + b (c) n (c) + c (c) p (c)

lim_{t \to c} p r (t) \times u (t)] = a (c) m (c) + b (c) n (c) + c (c) p (c)

Now taking R.H.S

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