Preeti K.

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Linear Algebra

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Question:

Calculate the dot product of a=(1,2,3) and b=(4,−5,6), where a and b are vectors. Do the vectors form an acute angle, right angle, or obtuse angle?

Preeti K.

Answer:

Using the component formula for the dot product of three-dimensional vectors, a⋅b = $$a_1{b_1} + a_2b_2 + a_3b_3 $$ a.b = $${1}.{4} + 2.(-5)+ 3.6 $$ a.b = $$4 - 10 + 18 = 22 - 10 = 12 $$ Since, a.b > 0, it implies that angle between two vectors is acute.

Geometry

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Question:

Find the value of each interior angle of an 11 sided polygon. Round the answer to the nearest hundredth.

Preeti K.

Answer:

The sum of the n-sided polygon is given as: $$Sum = (n-2)180^{\circ}$$ Now, One interior angle will be given as : Interior angle measure = $$\frac{(n-2)180^{\circ}}{n}$$ Now, in 11 sided polygon, there are 11 sides which means n=11 Put n=11, in the above equation to find each interior angle measure. Interior angle measure = $$\frac{(n-2)180^{\circ}}{n}$$ = $$\frac{(11-2)180^{\circ}}{11}$$ =$$\frac{(9)180^{\circ}}{11}$$ = $$147.27^{\circ}$$

Algebra

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Question:

Simplify the expression: $$\frac {x^{4}}{x^{3}}$$

Preeti K.

Answer:

Here, in the question, we have $$x^{4}$$ in the numerator and $$x^{3}$$ in the denominator. Since, the base is same which is x, we can apply exponent rule of division: $$\frac{a^{m}}{a^{n}} $$ = $$a^{m-n}$$ The exponents gets subtracted, if base is same when dividing. Therefore, $$\frac {x^{4}}{x^{3}}$$ = $$x^{4-3}$$ = $$x^{1}$$ = $$x$$

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