Question

A vector has an x component of -23.0 units and a y component of 35.8 units. Find the magnitude and direction of this vector.

Answer 1

Answer on Question 70109, Physics, Mechanics | Relativity

Question:

A vector has an $x$-component of $-23.0$ units and a $y$-component of $35.8$ units. Find the magnitude and direction of this vector.

Solution:

a) We can find the magnitude of the vector $\vec{a}$ from the formula:

$$|\vec{a}| = \sqrt{a_x^2 + a_y^2} = \sqrt{(-23.0)^2 + (35.8)^2} = 42.55 \text{ units}.$$

b) We can find the direction of the vector from the formula:

$$\alpha = \tan^{-1} \frac{a_y}{a_x} = \tan^{-1} \frac{35.8}{-23.0} = -57.3{}^\circ.$$

However, we must add $180{}^\circ$ to obtain the correct answer:

$$\alpha = -57.3{}^\circ + 180{}^\circ = 123{}^\circ.$$

The angle between the vector $\vec{a}$ and the positive $x$-axis is $123{}^\circ$.

Answer:

a) $|\vec{a}| = 42.55$ units.

b) $\alpha = 123{}^\circ$.

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