1.1: Vectors in the Plane and Space

A vector in a plane is represented by a directed line segment (an arrow). The endpoints of the segment are called the initial point and the terminal point of the vector. An arrow from the initial point to the terminal point indicates the direction of the vector. The length of the line segment represents its magnitude. We use the notation \(\|\vecs{v}\|\) to denote the magnitude of the vector \(\vecs{v}\). A vector with an initial point and terminal point that are the same is called the zero vector, denoted \(\vecs{0}\). The zero vector is the only vector without a fixed direction, and by convention can be considered to have any direction convenient to the problem at hand.

Vectors with the same magnitude and direction are called equivalent vectors. We treat equivalent vectors as equal, even if they have different initial points. Thus, if \(\vecs{v}\) and \(\vecs{w}\) are equivalent, we write

\[\vecs{v}=\vecs{w} \nonumber \]

The arrows in Figure \(\PageIndex{9 (b)}\) are equivalent. Each arrow has the same length and direction. A closely related concept is the idea of parallel vectors. Two vectors are said to be parallel if they have identical or opposite directions. We explore this idea in more detail later in the chapter. A vector is defined by its magnitude and direction, regardless of where its initial point is located.

The use of boldface, lowercase letters to name vectors is a common representation in print, but there are alternative notations. When writing the name of a vector by hand, for example, it is easier to sketch an arrow over the variable than to show it is a vector: \(\vecs{v}\). When a vector has initial point \(P\) and terminal point \(Q\), the notation \(\vecd{PQ}\) is useful because it indicates the direction and location of the vector.

Vectors have many real-life applications, including situations involving force or velocity. For example, consider the forces acting on a boat crossing a river. The boat’s motor generates a force in one direction, and the current of the river generates a force in another direction. Both forces are vectors. We must take both the magnitude and direction of each force into account if we want to know where the boat will go.

A second example that involves vectors is a quarterback throwing a football. The quarterback does not throw the ball parallel to the ground; instead, he aims up into the air. The velocity of his throw can be represented by a vector. If we know how hard he throws the ball (magnitude—in this case, speed), and the angle (direction), we can tell how far the ball will travel down the field.

A real number is often called a scalar in mathematics and physics. Unlike vectors, scalars are generally considered to have a magnitude only, but no direction. Multiplying a vector by a scalar changes the vector’s magnitude. This is called scalar multiplication. Note that changing the magnitude of a vector does not indicate a change in its direction. For example, wind blowing from north to south might increase or decrease in speed while maintaining its direction from north to south.

As you might expect, if \(k=−1\), we denote the product \(k\vecs{v}\) as

\[k\vecs{v}=(−1)\vecs{v}=−\vecs{v} \nonumber \]

Note that \(−\vecs{v}\) has the same magnitude as \(\vecs{v}\), but has the opposite direction.

Another operation we can perform on vectors is to add them together in vector addition, but because each vector may have its own direction, the process is different from adding two numbers. The most common graphical method for adding two vectors is to place the initial point of the second vector at the terminal point of the first, as in Figure \(\PageIndex{11a}\).

It is also appropriate here to discuss vector subtraction. We define \(\vecs{v}−\vecs{w}\) as \(\vecs{v}+(−\vecs{w})=\vecs{v}+(−1)\vecs{w}\). The vector \(\vecs{v}−\vecs{w}\) is called the vector difference. Graphically, the vector \(\vecs{v}−\vecs{w}\) is depicted by drawing a vector from the terminal point of \(\vecs{w}\) to the terminal point of \(\vecs{v}\).

In Figure \(\PageIndex{3 (a)}\), the initial point of \(\vecs{v}+\vecs{w}\) is the initial point of \(\vecs{v}\). The terminal point of \(\vecs{v}+\vecs{w}\) is the terminal point of \(\vecs{w}\). These three vectors form the sides of a triangle. It follows that the length of any one side is less than the sum of the lengths of the remaining sides. So we have

\[\|\vecs{v}+\vecs{w}\|≤\|\vecs{v}\|+\|\vecs{w}\|. \nonumber \]

This is known more generally as the triangle inequality. There is one case, however, when the resultant vector \(\vecs{v}+\vecs{w}\) has the same magnitude as the sum of the magnitudes of \(\vecs{v}\) and \(\vecs{w}\). This happens only when \(\vecs{v}\) and \(\vecs{w}\) have the same direction.

Working with vectors in a plane is easier when we are working in a coordinate system. When the initial points and terminal points of vectors are given in Cartesian coordinates, computations become straightforward.

We have seen how to plot a vector when we are given an initial point and a terminal point. However, because a vector can be placed anywhere in a plane, it may be easier to perform calculations with a vector when its initial point coincides with the origin. We call a vector with its initial point at the origin a standard-position vector. Because the initial point of any vector in standard position is known to be \((0,0)\), we can describe the vector by looking at the coordinates of its terminal point.

Recall that vectors are named with lowercase letters in bold type or by drawing an arrow over their name. We have also learned that we can name a vector by its component form, with the coordinates of its terminal point in angle brackets. However, when writing the component form of a vector, it is important to distinguish between \(⟨x,y⟩\) and \((x,y)\). The first ordered pair uses angle brackets to describe a vector, whereas the second uses parentheses to describe a point in a plane. The initial point of \(⟨x,y⟩\) is \((0,0)\); the terminal point of \(⟨x,y⟩\) is \((x,y)\).

When we have a vector not already in standard position, we can determine its component form in one of two ways. We can use a geometric approach, in which we sketch the vector in the coordinate plane, and then sketch an equivalent standard-position vector. Alternatively, we can find it algebraically, using the coordinates of the initial point and the terminal point. To find it algebraically, we subtract the \(x\)-coordinate of the initial point from the \(x\)-coordinate of the terminal point to get the \(x\)-component, and we subtract the \(y\)-coordinate of the initial point from the \(y\)-coordinate of the terminal point to get the \(y\)-component.

To find the magnitude of a vector, we calculate the distance between its initial point and its terminal point. The magnitude of vector \(\vecs{v}=⟨x,y⟩\) is denoted \(\|\vecs{v}\|,\) or \(|\vecs{v}|\), and can be computed using the formula

\[\|\vecs{v}\|=\sqrt{x^2+y^2} \nonumber \]

Note that because this vector is written in component form, it is equivalent to a vector in standard position, with its initial point at the origin and terminal point \((x,y)\). Thus, it suffices to calculate the magnitude of the vector in standard position. Using the distance formula to calculate the distance between initial point \((0,0)\) and terminal point \((x,y)\), we have

\[\|\vecs{v}\|=\sqrt{(x−0)^2+(y−0)^2}=\sqrt{x^2+y^2} \nonumber \]

Based on this formula, it is clear that for any vector \(\vecs{v}, \|\vecs{v}\|≥0,\) and \(\|\vecs{v}\|=0\) if and only if \(\vecs{v}=\vecs{0}\).

The magnitude of a vector can also be derived using the Pythagorean theorem, as in the following figure.

We have defined scalar multiplication and vector addition geometrically. Expressing vectors in component form allows us to perform these same operations algebraically.

Now that we have established the basic rules of vector arithmetic, we can state the properties of vector operations. We will prove two of these properties. The others can be proved in a similar manner.

We have found the components of a vector given its initial and terminal points. In some cases, we may only have the magnitude and direction of a vector, not the points. For these vectors, we can identify the horizontal and vertical components using trigonometry.

Consider the angle \(θ\) formed by the vector \(\vecs{v}\) and the positive \(x\)-axis. We can see from the triangle that the components of vector \(\vecs{v}\) are \(⟨\|\vecs{v}\| \cos(θ), \, \|\vecs{v}\| \sin (θ)⟩\). Therefore, given an angle and the magnitude of a vector, we can use the cosine and sine of the angle to find the components of the vector.

A unit vector is a vector with magnitude \(1\). For any nonzero vector \(\vecs{v}\), we can use scalar multiplication to find a unit vector \(\vecs{u}\) that has the same direction as \(\vecs{v}\). To do this, we multiply the vector by the reciprocal of its magnitude:

\[\vecs{u}=\dfrac{1}{\|\vecs{v}\|} \vecs{v} \nonumber \]

Recall that when we defined scalar multiplication, we noted that \(\|k\vecs{v}\| =|k|⋅\|\vecs{v}\| \). For \(\vecs{u}=\dfrac{1}{\|\vecs{v}\| }\vecs{v}\), it follows that \(\|\vecs{u}\| =\dfrac{1}{\|\vecs{v}\| }(\|\vecs{v}\| )=1\). We say that \(\vecs{u}\) is the unit vector in the direction of \(\vecs{v}\). The process of using scalar multiplication to find a unit vector with a given direction is called normalization.

We have seen how convenient it can be to write a vector in component form. Sometimes, though, it is more convenient to write a vector as a sum of a horizontal vector and a vertical vector. To make this easier, let’s look at standard unit vectors. The standard unit vectors are the vectors \(\hat{\mathbf i}=⟨1,0⟩\) and \(\hat{\mathbf j}=⟨0,1⟩\)

By applying the properties of vectors, it is possible to express any vector in terms of \(\hat{\mathbf i}\) and \(\hat{\mathbf j}\) in what we call a linear combination:

\[\vecs{v}=⟨x,y⟩=⟨x,0⟩+⟨0,y⟩=x⟨1,0⟩+y⟨0,1⟩=x\hat{\mathbf i}+y\hat{\mathbf j} \nonumber \]

Thus, \(\vecs{v}\) is the sum of a horizontal vector with magnitude \(x\), and a vertical vector with magnitude \(y\), as in Figure \(\PageIndex{17}\).

Just like two-dimensional vectors, three-dimensional vectors are quantities with both magnitude and direction, and they are represented by directed line segments (arrows). With a three-dimensional vector, we use a three-dimensional arrow.

Three-dimensional vectors can also be represented in component form. The notation \(\vecs{v}=⟨x,y,z⟩\) is a natural extension of the two-dimensional case, representing a vector with the initial point at the origin, \((0,0,0)\), and terminal point \((x,y,z)\). The zero vector is \(\vecs{0}=⟨0,0,0⟩\). So, for example, the three dimensional vector \(\vecs{v}=⟨2,4,1⟩\) is represented by a directed line segment from point \((0,0,0)\) to point \((2,4,1)\).

Vector addition and scalar multiplication are defined analogously to the two-dimensional case. If \(\vecs{v}=⟨x_1,y_1,z_1⟩\) and \(\vecs{w}=⟨x_2,y_2,z_2⟩\) are vectors, and \(k\) is a scalar, then

\[\vecs{v}+\vecs{w}=⟨x_1+x_2,y_1+y_2,z_1+z_2⟩ \nonumber \]

and

\[k\vecs{v}=⟨kx_1,ky_1,kz_1⟩ \nonumber \]

If \(k=−1,\) then \(k\vecs{v}=(−1)\vecs{v}\) is written as \(−\vecs{v}\), and vector subtraction is defined by \(\vecs{v}−\vecs{w}=\vecs{v}+(−\vecs{w})=\vecs{v}+(−1)\vecs{w}\).

The standard unit vectors extend easily into three dimensions as well, \(\hat{\mathbf i}=⟨1,0,0⟩\), \(\hat{\mathbf j}=⟨0,1,0⟩\), and \(\hat{\mathbf k}=⟨0,0,1⟩\), and we use them in the same way we used the standard unit vectors in two dimensions. Thus, we can represent a vector in \(ℝ^3\) in the following ways:

\[\vecs{v}=⟨x,y,z⟩=x\hat{\mathbf i}+y\hat{\mathbf j}+z\hat{\mathbf k} \nonumber \]

As described earlier, vectors in three dimensions behave in the same way as vectors in a plane. The geometric interpretation of vector addition, for example, is the same in both two- and three-dimensional space.

We have already seen how some of the algebraic properties of vectors, such as vector addition and scalar multiplication, can be extended to three dimensions. Other properties can be extended in similar fashion. They are summarized here for our reference.

We have seen that vector addition in two dimensions satisfies the commutative, associative, and additive inverse properties. These properties of vector operations are valid for three-dimensional vectors as well. Scalar multiplication of vectors satisfies the distributive property, and the zero vector acts as an additive identity. The proofs to verify these properties in three dimensions are straightforward extensions of the proofs in two dimensions.

---