3. Vectors

A velocity vector 40 degrees below the positive x-axis has a y-component of -10 m/s. What is the value of its x-component?

The bacterium E. coli is a single-cell organism that lives in the gut of healthy animals, including humans. When grown in a uniform medium in the laboratory, these bacteria swim along zig-zag paths at a constant speed of 20 μm/s. FIGURE P3.42 shows the trajectory of an E. coli as it moves from point A to point E. What are the magnitude and direction of the bacterium's average velocity for the entire trip?

FIGURE P3.26 shows vectors A and B. Find D = 2A ＋B Write your answer in component form.

Ruth sets out to visit her friend Ward, who lives 50 mi north and 100 mi east of her. She starts by driving east, but after 30 mi she comes to a detour that takes her 15 mi south before going east again. She then drives east for 8 mi and runs out of gas, so Ward flies there in his small plane to get her. What is Ward's displacement vector? Give your answer (a) in component form, using a coordinate system in which the y-axis points north, and (b) as a magnitude and direction.

While vacationing in the mountains you do some hiking. In the morning, your displacement is $\mathbf{S}_{morning} = (2000 \, \text{m}, \text{east}) + (3000 \, \text{m}, \text{north}) + (200 \, \text{m}, \text{vertical})$. Continuing on after lunch, your displacement is $\mathbf{S}_{afternoon} = (1500 \, \text{m}, \text{west}) + (2000 \, \text{m}, \text{north}) - (300 \, \text{m}, \text{vertical})$. What is the magnitude of your net displacement for the day?

While vacationing in the mountains you do some hiking. In the morning, your displacement is $S_{morning}=(2000\text{m},\text{east})+(3000\text{m},\text{north})+(200\text{m},\text{vertical})$. Continuing on after lunch, your displacement is $S_{afternoon}=(1500\text{m},\text{west})+(2000\text{m},\text{north})-(300\text{m},\text{vertical})$. At the end of the hike, how much higher or lower are you compared to your starting point?

A runner is training for an upcoming marathon by running around a 100-m-diameter circular track at constant speed. Let a coordinate system have its origin at the center of the circle with the x-axis pointing east and the y-axis north. The runner starts at (x,y) = (50m, 0m) and runs 2.5 times around the track in aclockwise direction. What is his displacement vector? Give your answer as a magnitude and direction.

A crate, seen from above, is pulled with three ropes that have the tensions shown in FIGURE P3.44. Tension is a vector directed along the rope, measured in newtons (abbreviated N). Suppose the three ropes are replaced with a single rope that has exactly the same effect on the crate. What is the tension in this rope? Write your answer in component form using unit vectors.

Draw each of the following vectors, label an angle that specifies the vector's direction, then find its magnitude and direction. r = (-2.0i - 1.0j) cm

Draw each of the following vectors, label an angle that specifies the vector's direction, then find its magnitude and direction. B = -4.0i + 4.0j

Draw each of the following vectors, label an angle that specifies the vector's direction, and then find the vector's magnitude and direction. v = (14i - 11j) m/s

Draw each of the following vectors, label an angle that specifies the vector's direction, and then find the vector's magnitude and direction. A = 3.0i + 7.0j

Let $A=(3.0\text{m},{20}^{\circ }\text{south of east}),B=(2.0\text{m},\text{north})$, and $C=(5.0\text{m},{70}^{\circ }\text{south of west})$. Find the magnitude and the direction of $D=A+B+C$.

Let A = 4i - 2j, B = -3i + 5j, and F = A - 4B. Write vector F in component form.

Let A = 4i - 2j, B = -3i + 5j, and E = 2A + 3B. What are the magnitude and direction of vector E?