Question

Let \(\mathbf{u}=\langle 3,-4\rangle, \mathbf{v}=\langle 1,1\rangle,\) and \(\mathbf{w}=\langle 1,0\rangle\) Find two vectors parallel to \(\mathbf{u}\) with four times the magnitude of \(\mathbf{u}\).

Short answer

Question: Find two vectors parallel to the vector u = <3, -4> with four times the magnitude of u. Answer: The two vectors parallel to u = <3, -4> with four times the magnitude of u are p = <12, -16> and q = <-12, 16>.

Step-by-step solution

Calculate the scalar multiple of the vector

Multiply the given vector \(\mathbf{u}=\langle 3,-4\rangle\) by the scalar \(4\) to get the first vector that is parallel to \(\mathbf{u}\) and has four times its magnitude: \(\mathbf{p} = 4\mathbf{u} = 4\langle 3,-4\rangle = \langle 12,-16\rangle\) So, \(\mathbf{p}=\langle 12,-16\rangle\) is the first vector that is parallel to \(\mathbf{u}\) and has four times its magnitude.

Calculate the second parallel vector

Now, we will find the second vector. This can be achieved by multiplying the given vector \(\mathbf{u}\) by \(-4\). This will result in a vector that is parallel to \(\mathbf{u}\) but in the opposite direction: \(\mathbf{q} = -4\mathbf{u} = -4\langle 3,-4\rangle = \langle -12,16\rangle\) This means the second vector parallel to \(\mathbf{u}\) with the same magnitude as \(\mathbf{p}\) but in the opposite direction is \(\mathbf{q}=\langle -12,16\rangle\).

Present the final answer

The two vectors parallel to \(\mathbf{u}=\langle 3,-4\rangle\) with four times the magnitude of \(\mathbf{u}\) are: \(\mathbf{p}=\langle 12,-16\rangle\) and \(\mathbf{q}=\langle -12,16\rangle\)

Key concepts

Scalar Multiplication

In mathematics, scalar multiplication refers to the operation of multiplying a vector by a real number, known as a scalar. This operation alters both the magnitude and direction of the original vector. For instance, when we multiply a vector, such as \(\boldsymbol{u} = \langle 3, -4 \rangle\), by a scalar like \(4\), the resulting vector \(\boldsymbol{p} = 4\boldsymbol{u}\) will have four times the length but the same direction as the original vector.

However, if we use a negative scalar, such as \(-4\), the resulting vector \(\boldsymbol{q} = -4\boldsymbol{u}\) will have the same magnitude as \(\boldsymbol{p}\) but will point in the opposite direction. This is crucial in understanding vectors in physics and engineering, where we often scale vectors to represent quantities such as force or velocity.

Vectors in Mathematics

Vectors are fundamental in mathematics and represent quantities that have both magnitude (size) and direction. They can be depicted visually as arrows on a graph, with the arrow's length corresponding to the vector's magnitude and the arrow's direction showing the way the vector points. Vectors can be expressed in a coordinate system using ordered pairs, such as \(\boldsymbol{v} = \langle 1, 1 \rangle\), where the first number represents the horizontal component and the second number the vertical component.

Understanding vectors is indispensable in fields like physics, where they represent a myriad of phenomena, from velocity to acceleration and forces. They are also the cornerstone of linear algebra and calculus, instrumental in describing lines, planes, and transformations within space.

Magnitude of a Vector

The magnitude of a vector, often called its length, is a measure of how long the vector is. We calculate it using the Pythagorean theorem for the two-dimensional vector \(\boldsymbol{u} = \langle a, b \rangle\) by taking the square root of the sum of the squared components: the formula is \(|\boldsymbol{u}| = \sqrt{a^2 + b^2}\).

For the vector \(\boldsymbol{u} = \langle 3, -4 \rangle\), the magnitude would be \(|\boldsymbol{u}| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\). The magnitude is a scalar quantity and carries no directional information, meaning it is always a positive number or zero.

Direction of a Vector

The direction of a vector is defined by the angle it makes with a reference axis, usually the positive x-axis in the Cartesian coordinate system. In two dimensions, for a vector \(\boldsymbol{v} = \langle a, b \rangle\), the direction is determined by the slope of the line that extends from the origin to the point \((a, b)\).

We can calculate the direction using trigonometry, specifically the tangent function, which relates the ratio of the opposite side to the adjacent side of a right triangle. The direction can also be visually assessed in a graph by examining the arrow representing the vector; it points towards the vector's direction. It's important to note that two vectors with the same direction will be parallel, even if they have different magnitudes.