Vector Basics: Learn It 2

Giselle Miller

Vector Basics: Learn It 2

Combining Vectors

Vectors appear everywhere in real-world situations involving forces, velocities, and other directional quantities. Understanding how to combine and manipulate vectors is essential for solving these problems.

Consider a boat crossing a river. The boat’s motor creates a force in one direction, while the river current creates a force in another direction. To predict where the boat will actually go, you need to account for both forces—their magnitudes and directions.

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.

Before working with vector operations, let’s clarify an important distinction. A scalar is a real number that has only magnitude—like temperature, mass, or speed. Unlike vectors, scalars have no direction component.

Scalar vs. Vector:

Scalar: Has magnitude only (examples: [latex]5[/latex], [latex]-3.2[/latex], [latex]\pi[/latex])
Vector: Has both magnitude and direction (examples: velocity, force, displacement)

Scalar Multiplication

When you multiply a vector by a scalar, you change the vector’s magnitude but affect its direction in a predictable way. This operation is called scalar multiplication.

scalar multiplication

For a vector [latex]\mathbf{v}[/latex] and scalar [latex]k[/latex], the product [latex]k\mathbf{v}[/latex] is a vector where:

Magnitude: [latex]|k|[/latex] times the magnitude of [latex]\mathbf{v}[/latex]
Direction: Same as [latex]\mathbf{v}[/latex] if [latex]k > 0[/latex], opposite if [latex]k < 0[/latex]
If [latex]k = 0[/latex] or [latex]\mathbf{v} = \mathbf{0}[/latex], then [latex]k\mathbf{v} = \mathbf{0}[/latex]

Think of scalar multiplication as stretching or shrinking a vector. If [latex]k = 2[/latex], you double the vector’s length. If [latex]k = 0.5[/latex], you halve it. If [latex]k = -1[/latex], you flip the vector to point in the opposite direction without changing its length.

When [latex]k = -1[/latex], we write the result as:

[latex]k\mathbf{v} = (-1)\mathbf{v} = -\mathbf{v}[/latex]

The vector [latex]-\mathbf{v}[/latex] has the same magnitude as [latex]\mathbf{v}[/latex] but points in the opposite direction.

This graphic has 4 figures. The first figure is a vector labeled “v.” The second figure is a vector twice as long as the first vector and is labeled “2 v.” The third figure is half as long as the first and is labeled “1/2 v.” The fourth figure is a vector in the opposite direction as the first. It is labeled “-v.” — Figure 4. (a) The original vector [latex]{\bf{v}}[/latex] has length [latex]n[/latex] units. (b) The length of [latex]2{\bf{v}}[/latex] equals [latex]2n[/latex] units. (c) The length of [latex]{\bf{v}}{/}2[/latex] is [latex]n{/}2[/latex] units. (d) The vectors [latex]{\bf{v}}[/latex] and [latex]-{\bf{v}}[/latex] have the same length but opposite directions.

Wind blowing north at [latex]15[/latex] mph can be represented by vector [latex]\mathbf{v}[/latex]. Then:

[latex]2\mathbf{v}[/latex] represents wind blowing north at [latex]30[/latex] mph
[latex]0.5\mathbf{v}[/latex] represents wind blowing north at [latex]7.5[/latex] mph
[latex]-\mathbf{v}[/latex] represents wind blowing south at [latex]15[/latex] mph

Vector Addition

Adding vectors is different from adding regular numbers because each vector has its own direction. You need graphical methods to combine them properly.

Triangle Method

The most common approach places the initial point of the second vector at the terminal point of the first vector.

Think about this in terms of displacement. If you walk from point [latex]A[/latex] to point [latex]B[/latex] (following vector [latex]\mathbf{v}[/latex]), then walk from point [latex]B[/latex] to point [latex]C[/latex] (following vector [latex]\mathbf{w}[/latex]), your total displacement is the same as walking directly from [latex]A[/latex] to [latex]C[/latex]. This direct path represents the vector sum [latex]\mathbf{v} + \mathbf{w}[/latex].

The order doesn’t matter—you get the same result whether you add [latex]\mathbf{v} + \mathbf{w}[/latex] or [latex]\mathbf{w} + \mathbf{v}[/latex]. This is called the commutative property of vector addition.

Notice that the three vectors [latex]\mathbf{v}[/latex], [latex]\mathbf{w}[/latex], and [latex]\mathbf{v} + \mathbf{w}[/latex] form a triangle. From basic geometry, the length of any side of a triangle must be less than or equal to the sum of the other two sides. This gives us the triangle inequality:

[latex]||\mathbf{v} + \mathbf{w}|| \leq ||\mathbf{v}|| + ||\mathbf{w}||[/latex]

The only time we get equality is when [latex]\mathbf{v}[/latex] and [latex]\mathbf{w}[/latex] point in exactly the same direction.

Parallelogram Method

An alternative approach places both vectors at the same initial point, then draws a parallelogram using the vectors as adjacent sides. The diagonal of the parallelogram represents the vector sum.

Both methods give identical results—use whichever seems clearer for your problem.

This image has two figures. The first has two vectors, v and w with the same initial point. A parallelogram is formed by sketching broken lines parallel to the two vectors. A diagonal line is drawn from the same initial point to the opposite corner. It is labeled “v + w.” The second has two vectors, v and w. Vector v begins at the terminal point of vector w. A parallelogram is formed by sketching broken lines parallel to the two vectors. A diagonal line is drawn from the same initial point as vector w to the opposite corner. It is labeled “v + w.” — Figure 5. (a) When adding vectors by the triangle method, the initial point of [latex]{\bf{w}}[/latex] is the terminal point of [latex]{\bf{v}}[/latex]. (b) When adding vectors by the parallelogram method, the vectors [latex]{\bf{v}}[/latex] and [latex]{\bf{w}}[/latex] have the same initial point.

vector addition

To add vectors [latex]\mathbf{v}[/latex] and [latex]\mathbf{w}[/latex] graphically, place the initial point of [latex]\mathbf{w}[/latex] at the terminal point of [latex]\mathbf{v}[/latex]. The vector sum [latex]\mathbf{v} + \mathbf{w}[/latex] runs from the initial point of [latex]\mathbf{v}[/latex] to the terminal point of [latex]\mathbf{w}[/latex].

Vector Subtraction

Vector subtraction uses the same principle as adding the negative of a vector. We define:

[latex]\mathbf{v} - \mathbf{w} = \mathbf{v} + (-\mathbf{w})[/latex]

Graphically, [latex]\mathbf{v} - \mathbf{w}[/latex] is the vector that points from the terminal point of [latex]\mathbf{w}[/latex] to the terminal point of [latex]\mathbf{v}[/latex].

This image has two figures. The first figure has two vectors, one labeled “v” and the other labeled “w.” Both vectors have the same initial point. A third vector is drawn between the terminal points of v and w. It is labeled “v – w.” The second figure has two vectors, one labeled “v” and the other labeled “-w.” The vector “-w” has its initial point at the terminal point of “v.” A parallelogram is created with broken lines where “v” is the diagonal and “w” is the top side. — Figure 6. (a) The vector difference [latex]{\bf{v-w}}[/latex] is depicted by drawing a vector from the terminal point of [latex]{\bf{w}}[/latex] to the terminal point of [latex]{\bf{v}}[/latex]. (b) The vector [latex]{\bf{v-w}}[/latex] is equivalent to the vector [latex]{\bf{v+(-w)}}[/latex].

In Figure 4(a), the initial point of [latex]\bf{v + w}[/latex] is the initial point of [latex]\bf{v}[/latex]. The terminal point of [latex]\bf{v + w}[/latex] is the terminal point of [latex]\bf{w}[/latex] . 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

[latex]\bf{|| v + w|| \leq ||v|| + ||w||}[/latex].

This is known more generally as the triangle inequality. There is one case, however, when the resultant vector [latex]\bf{u + v}[/latex] has the same magnitude as the sum of the magnitudes of [latex]\bf{u}[/latex] and [latex]\bf{v}[/latex]. This happens only when [latex]\bf{u}[/latex] and [latex]\bf{v}[/latex] have the same direction.

Given the vectors [latex]\bf{v}[/latex] and [latex]\bf{w}[/latex] shown in Figure 6, sketch the vectors.

[latex]\bf{3w}[/latex]
[latex]\bf{v + w}[/latex]
[latex]\bf{2v - w}[/latex]

This figure has two vectors. They are vector v and vector w. They are not connected. — Figure 7. Vectors [latex]{\bf{v}}[/latex] and [latex]{\bf{w}}[/latex] lie in the same plane.