WEBVTT
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Find the angle between the vectors 𝐮 three, negative two and 𝐯 negative five, negative three.
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Give your answer to one decimal place.
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In order to answer this question, we recall that the cosine of the angle between two vectors is equal to the dot product of two vectors 𝐮 and 𝐯 divided by the magnitude of vector 𝐮 multiplied by the magnitude of vector 𝐯.
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Let’s begin by calculating the dot product of vector 𝐮 and vector 𝐯.
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We multiply the 𝑥- and 𝑦-components separately and then find the sum of these values.
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In this question, the 𝑥-components are three and negative five.
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The 𝑦-components are negative two and negative three.
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Multiplying a positive number and a negative number gives a negative answer.
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Therefore, three multiplied by negative five is negative 15.
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Multiplying two negative numbers in this case negative two and negative three gives us positive six.
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The dot product of vectors 𝐮 and 𝐯 is, therefore, equal to negative nine.
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To find the magnitude of any vector, we find the sum of the squares of its individual components and then square root our answer.
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The magnitude of vector 𝐮 is equal to the square root of three squared plus negative two squared.
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Three squared is equal to nine.
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And negative two squared is equal to four.
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As nine plus four equals 13, the magnitude of vector 𝐮 is the square root of 13.
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We repeat this to calculate the magnitude of vector 𝐯.
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Negative five squared is 25, and negative three squared is nine.
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These have a sum of 34.
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Therefore, the magnitude of vector 𝐯 is equal to the square root of 34.
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Substituting in our values, we see that the cos of angle 𝜃 is equal to negative nine over the square root of 13 multiplied by the square root of 34.
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We can then calculate the angle 𝜃 by taking the inverse cosine of both sides of this equation.
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Typing this into our calculator, we get 𝜃 is equal to 115.346 and so on.
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We are asked to round our answer correct to one decimal place.
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As the deciding number in the hundredths column is less than five, we will round down.
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The angle between the two vectors 𝐮 and 𝐯 is equal to 115.3 degrees.