Question

Three physical quantities are given by the following vectors: !
F1 = (4.0ˆi + 6.0 ˆj) N
F2 = (2.5ˆi − 5.0 ˆj) N
r = (4.0ˆi + 4.0 ˆj) m
a) What is the magnitude and direction of the vector F1?
b) Draw a diagram showing the vectors F1 and F2,
together with the vectors F1 + F2 and F1 − F2.
c) Find the quantity Fr = r . F1.
d) Find the vector τ = r × F1.

Answer 1

Answer on Question #53952, Physics / Other

Three physical quantities are given by the following vectors:

$$F _ {1} = (4. 0 ^ {\wedge} i + 6. 0 ^ {\wedge} j) N$$ $$F _ {2} = (2. 5 ^ {\wedge} i - 5. 0 ^ {\wedge} j) N$$ $$r = (4. 0 ^ {\wedge} i + 4. 0 ^ {\wedge} j) m$$

a) What is the magnitude and direction of the vector $F_{1}$ ?

b) Draw a diagram showing the vectors $F_{1}$ and $F_{2}$ , together with the vectors $F_{1} + F_{2}$ and $F_{1} - F_{2}$ .

c) Find the quantity $F_{r} = r \cdot F_{1}$ .

d) Find the vector $\tau = r\times F_1$

Solution:

a) What is the magnitude and direction of the vector $F_{1}$ ?

The magnitude is

$$\left| F _ {1} \right| = \sqrt {x ^ {2} + y ^ {2}} = \sqrt {4 ^ {2} + 6 ^ {2}} = 2 \sqrt {1 3} = 7. 2 1$$

To find direction the following formula can be used:

$$\tan \theta = \frac {y}{x} = \frac {6}{4} = 1. 5$$ $$\theta = \tan^ {- 1} (1. 5) = 5 6. 3 1 {}^ {\circ}$$

Answer: a) $F_{1} = 7.21 \, \text{N}$ at $56.31{}^{\circ}$ above the positive x-axis

b) Draw a diagram showing the vectors $F_{1}$ and $F_{2}$ , together with the vectors $F_{1} + F_{2}$ and $F_{1} - F_{2}$ .

c) Find the quantity $F_{r} = r \cdot F_{1}$ .

You can calculate the Dot Product of two vectors this way:

$$a \cdot b = a _ {x} * b _ {x} + a _ {y} * b _ {y}$$

In our case

$$F _ {r} = 4 * 4 + 4 * 6 = 4 0$$

Answer: c) $F_{r} = 40$

d) Find the vector $\tau = r\times F_1$

The Cross Product $a \times b$ of two vectors is another vector that is at right angles to both:

And it all happens in 3 dimensions!

When a and b start at the origin point $(0,0,0)$ , the Cross Product will end at:

$$c _ {x} = a _ {y} b _ {z} - a _ {z} b _ {y}$$ $$c _ {y} = a _ {z} b _ {x} - a _ {x} b _ {z}$$ $$c _ {z} = a _ {x} b _ {y} - a _ {y} b _ {x}$$

In our case the cross product is

$$\tau = r \times F _ {1}$$ $$c _ {x} = 4 * 0 - 0 * 6 = 0$$ $$c _ {y} = 0 * 4 - 4 * 0 = 0$$ $$c _ {z} = 4 * 6 - 4 * 4 = 8$$

Answer: d) $\tau = 8\hat{k}$ Nm

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