A boat’s velocity, measured in meters per second, is described by vector = 〈 -3, 4 〉. In two or more complete sentences explain how to find the speed of the boat and the direction it is traveling in standard position. In your final answer, include all of your calculations.

Velocity is a vector, so it has a magnitude and direction.

Speed is its magnitude, which is obtained from

|〈-3, 4〉 m/s| = √((-3 m/s)² + (4 m/s)²) = √(25 m²/s²) = 5 m/s

Its direction is an angle θ made with the positive horizontal axis, satisfying

tan(θ) = (4 m/s) / (-3 m/s) = -4/3   →   θ = arctan(-4/3) + 180°n

where n is any integer. Now you have to consider that the x coordinate is negative and the y coordinate is positive, so 〈-3, 4〉 points into the second quadrant, and we get an angle there for n = 1 of about

θ ≈ 126.87°

Alternatively, you can use the vector 〈1, 0〉 in place of the axis, then compute the angle by relating it to the dot product, so θ is such that

〈-3, 4〉 • 〈1, 0〉 = |〈-3, 4〉| |〈1, 0〉| cos(θ)

(-3)•1 + 4•0 = √((-3)² + 4²) • √(1² + 0²) cos(θ)

-3/5 = cos(θ)   →   θ = arccos(-3/5) + 360°n

where again, n is any integer, and we get the same solution θ ≈ 126.87° in the second quadrant when n = 0.

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step-by-step explanation: