Find the terminal point of each of the following vectors. Draw each one as a directed line segment in the [math]xy[/math]-plane, and compute its length.

• [a [math]\vec v = (-3,4)_P[/math], where [math]P = (1,0)[/math],
• [math]\vec u = (4, -3)_P[/math], where [math]P = (1,0)[/math],
• [math]\vec x = (3,0)_Q[/math], where [math]Q = (-1,-1)[/math],
• [math]\vec a = (4\frac12,3\frac12)_O[/math], where [math]O = (0,0)[/math].

Let [math]P = (2,1)[/math]. Compute the terminal point of each of the following vectors, and draw each one as an arrow in the [math]xy[/math]-plane. The vectors [math]\vec u[/math] and [math]\vec v[/math] in parts \ref{ex10.3.2b}, \ref{ex10.3.2c}, \ref{ex10.3.2d}, and \ref{ex10.3.2e} are defined as in part \ref{ex10.3.2a}.

• lab{10.3.2a} [math]\vec u = (3,-2)_P[/math] and [math]\vec v = (1,1)_P[/math]
• lab{10.3.2b} [math]\vec u + \vec v[/math]
• lab{10.3.2c} [math]\vec u - \vec v[/math]
• lab{10.3.2d} [math]3\vec v[/math]
• lab{10.3.2e} [math]\vec u + 3\vec v[/math].

Let [math]P = (0,1)[/math], and consider the vectors [math]\vec x = (2,5)_P[/math] and [math]\vec y = (1,1)_P[/math].

• Draw the vectors [math]\vec x[/math], [math]\vec y[/math], and [math]\vec x + \vec y[/math] in the [math]xy[/math]-plane.
• Compute the lengths [math]|\vec x|[/math], [math]|\vec y|[/math], and [math]|\vec x+\vec y|[/math].

True or false: If [math]P \ne Q[/math], then [math]V_P[/math] and [math]V_Q[/math] are disjoint sets?

Let [math]\vec v[/math] be a vector in the plane with initial point [math]P[/math], and let [math]\theta[/math] be the angle whose vertex is [math]P[/math], whose initial side is the vector [math](1,0)_P)[/math], and whose terminal side is [math]\vec v[/math]. Show that

The angle [math]\theta[/math] is called the direction of the vector [math]\vec v[/math].

In physics, the force acting on a particle located at a point [math]P[/math] in the plane is represented by a vector. The length of the vector is the magnitude of the force (e.g., the number of pounds), and the direction of the vector is the direction of the force (see Problem Exercise).

• lab{10.3.6a} Draw the vector representing a force of [math]5[/math] pounds acting on a particle at the point [math](3,2)[/math] in a direction of [math]\frac{\pi}6[/math] radians.
• What are the coordinates of the force vector in \ref{ex10.3.6a}?

If a particle located at a point [math]P[/math] is simultaneously acted on by two forces [math]\vec u[/math] and [math]\vec v[/math], then the resultant force is the vector sum [math]\vec u + \vec v[/math]. The fact that vectors are added geometrically by constructing a parallelogram implies a corresponding Parallelogram Law of Forces. Suppose that a particle at the point [math](1,1)[/math] is simultaneously acted on by a force [math]\vec v[/math] of [math]10[/math] pounds in the direction of [math]\frac\pi6[/math] radians and a force [math]\vec u[/math] of [math]\sqrt{32}[/math] pounds in the direction of [math]-\frac\pi4[/math] radians.

• Draw the parallelogram of forces, and show the resultant force.
• What are the coordinates of the resultant force on the particle?

Addition and scalar multiplication are defined in the set [math]\R^2[/math] of all ordered pairs of real numbers by the equations

Show that [math]\R^2[/math] is a vector space with respect to these operations. This fact shows that the elements of a vector space need not necessarily be interpreted as arrows. The principal interpretation of [math]\R^2[/math] is that of the set of points of the plane.

True or false?

• The set [math]\R[/math] of all real numbers is a vector space with respect to ordinary addition and multiplication.
• The set [math]\C[/math] of all complex numbers is a vector space with respect to addition and multiplication by real numbers.
• The set [math]V[/math] of all vectors in the plane is a vector space with respect to vector addition and scalar multiplication as defined in this section.