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3 years ago

Determine the component form of the vector which has a magnitude of 30 & a direction of 116 degrees.

Mathematics

1 answer:

makkiz [27]3 years ago

6 0

Given :

Magnitude of vector is 30 units.

Direction of vector from x-axis is 116°.

To Find :

The component form of the vector.

Solution :

We know, vector in component form is given by :

$\vec{v} = (Acos\ \theta) \hat{i} + (Asin\ \theta)\hat{j}$

Putting given values,

$\vec{v} = (30\times cos\ 116^o) \hat{i} + (30\times sin\ 116^o)\hat{j}\\\\\vec{v} = -13.15 \hat{i} + 0.90 \hat{j}$

Hence, this is the required solution.

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In 1-4, to determine whether a sequence is either arithmetic or geometric, you need to look at differences of consecutive terms (arithmetic) and ratios of consecutive terms (geometric). If you can't find it, the sequence will fall under the "neither" category.

For example, the differences between consecutive terms in the first sequence are

$\left\{2-4,\dfrac12-2,\dfrac14-\dfrac12,\ldots\right\}=\left\{-2,-\dfrac32,-\dfrac14,\ldots\right\}$

If the sequence was arithmetic, the difference between consecutive terms would have been the same constant throughout this list. But that's not the case, so this sequence is not arithmetic.

The ratios between consecutive terms are

$\left\{\dfrac24,\dfrac{\frac12}2,\dfrac{\frac14}{\frac12},\ldots\right\}=\left\{\dfrac12,\dfrac14,\dfrac12,\ldots\right\}$

The sequence would have been geometric if the list contained the same value throughout, but it doesn't. So this sequence is neither arithmetic nor geometric.

Meanwhile, in the second sequence, the differences are

$\{-1-(-6),4-(-1),9-4,\ldots\}=\{5,5,5,\ldots\}$

so this sequence is arithmetic.

In 5-6, you know the sequences are arithmetic, so you know that they follow the recursive rule

$a_n=a_{n-1}+d$

For example, in the fifth sequence we know the first term is $a_1=4$ . The common difference between terms is $d=9-4=5$ . So using the rule above, we have the pattern