Question

Find the following vectors. The vector in the direction of \langle 5,-12\rangle with length 3

Short answer

Question: Given the vector \langle 5,-12\rangle, find a vector with a length of 3 in the same direction. Answer: The vector with a length of 3 in the same direction as \langle 5,-12\rangle is \left\langle\frac{15}{13},\frac{-36}{13}\right\rangle.

Step-by-step solution

Find the magnitude of the given vector

Determine the magnitude of the given vector \langle 5,-12\rangle by using the formula: Magnitude = \sqrt{(5)^2+(-12)^2} = \sqrt{25+144} = \sqrt{169} = 13

Find the unit vector

Now that we know the magnitude of the given vector, we can find the unit vector by dividing each component of the original vector by the magnitude: Unit vector = \left\langle\frac{5}{13}, \frac{-12}{13}\right\rangle

Adjust the length of the unit vector

Finally, we'll multiply each component of the unit vector by the desired length of 3: New vector = \left\langle 3\cdot\frac{5}{13}, 3\cdot\frac{-12}{13}\right\rangle = \left\langle\frac{15}{13},\frac{-36}{13}\right\rangle So, the vector in the direction of \langle 5,-12\rangle with the length of 3 is \left\langle \frac{15}{13},\frac{-36}{13}\right\rangle.