Question Video: Using the Relation between Two Vectors to Find the Value of an Unknown Symbol in Coordinates of Vectors Mathematics

For what value of 𝑘 are vectors 𝚨 = 〈7, −7𝑘, −6〉 and 𝚩 = 〈7, −3, 𝑘〉 perpendicular?

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Video Transcript

For what value of 𝑘 are vectors 𝚨 seven, negative seven 𝑘, negative six and 𝚩 seven, negative three, 𝑘 perpendicular?

We recall that the dot product of any two vectors 𝚨 and 𝚩 is equal to the magnitude of vector 𝚨 multiplied by the magnitude of vector 𝚩 multiplied by the cos of 𝜃, where 𝜃 is the angle between the two vectors.

If two vectors are perpendicular, as in this question, the angle between them will be equal to 90 degrees. We know that the cos of 90 degrees is equal to zero. This leads us to the fact that if two vectors are perpendicular, their dot product is equal to zero.

To find the dot products of two vectors in three dimensions, we find the product of their corresponding components and then the sum of these three values. Seven multiplied by seven is equal to 49. Multiplying negative seven 𝑘 by negative three gives us positive 21𝑘, as multiplying two negatives gives a positive answer. Negative six multiplied by 𝑘 is equal to negative six 𝑘. And adding this is the same as subtracting six 𝑘.

This gives us the equation zero is equal to 49 plus 21𝑘 minus six 𝑘. Subtracting 49 from both sides and collecting like terms gives us negative 49 is equal to 15𝑘. Finally, dividing both sides of this equation by 15 gives us 𝑘 is equal to negative 49 over 15. This is the value of 𝑘 such that vectors 𝚨 and 𝚩 are perpendicular.

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