3. Vectors

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.

FIGURE P3.43 shows three ropes tied together in a knot. One of your friends pulls on a rope with 3.0 units of force and another pulls on a second rope with 5.0 units of force. How hard and in what direction must you pull on the third rope to keep the knot from moving? Give the direction as an angle below the negative x-axis.

(II) For the vectors shown in Fig. 3–41, determine

(b) 2 A (→ above A) ― 3 B (→ above B) + 2 C (→ above C) , and

A position vector in the first quadrant has an x-component of 6 m and a magnitude of 10 m. What is the value of its y-component?

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

If two velocity vectors are added using the component method, which of the following correctly describes how to find the magnitude of the resultant velocity vector $v_r$ given their components $v_{1x}$, $v_{1y}$, $v_{2x}$, and $v_{2y}$?

Given two vectors $A$ and $B$ with components $A=(3,4)$ and $B=(1,-2)$, what are the components of the vector $A+B$?

Given two vectors in a plane, how many distinct combinations (regardless of magnitude or direction) can sum to produce an arbitrary third vector?

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.

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 $D=A+B+C$.

Which of the following does not describe the process of adding vectors by components?

Suppose you have four vectors of equal magnitude: $\mathrm{Vector} A$ points along the positive $x$-axis, $\mathrm{Vector} B$ points along the positive $y$-axis, $\mathrm{Vector} C$ points at $45°$ above the $x$-axis (in the first quadrant), and $\mathrm{Vector} D$ points at $45°$ below the $x$-axis (in the fourth quadrant). Which pair of vectors, when added, will result in the largest positive $y$ component?

Given two vectors $A$ and $B$ with components $A=\left(3,4\right)$ and $B=\left(1,-,2\right)$, what is the single vector that represents $A+B$ when added by components?

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.