Problem 26 - A Graphical Approach to Precalculus with Limits

Question

Given vectors u and v, find (a) \(2 u\) (b) \(2 u+3 v\) (c) \(v-3 u\) Do not use a calculator. $$\mathbf{u}=\langle- 2,-1\rangle, \mathbf{v}=\langle- 3,2\rangle$$

Step-by-Step Solution

Answer

(a) (-4, -2) (b) (-13, 4) (c) (3, 5)

Step 1: Find 2u

To find the vector \(2\mathbf{u}\), multiply each component of \(\mathbf{u}\) by 2. Given \(\mathbf{u} = \langle -2, -1 \rangle\), calculate: \[2\mathbf{u} = 2 \times (-2, -1) = (-4, -2)\]

Step 2: Find 2u + 3v

To find the vector \(2\mathbf{u} + 3\mathbf{v}\), first calculate \(3\mathbf{v}\) by multiplying each component of \(\mathbf{v}\) by 3. Given \(\mathbf{v} = \langle -3, 2 \rangle\), calculate: \[3\mathbf{v} = 3 \times (-3, 2) = (-9, 6)\]Now add the vectors \(2\mathbf{u}\) and \(3\mathbf{v}\):\[2\mathbf{u} + 3\mathbf{v} = (-4, -2) + (-9, 6) = (-13, 4)\]

Step 3: Find v - 3u

To find the vector \(\mathbf{v} - 3\mathbf{u}\), first calculate \(3\mathbf{u}\) by multiplying each component of \(\mathbf{u}\) by 3. \[3\mathbf{u} = 3 \times (-2, -1) = (-6, -3)\]Now subtract \(3\mathbf{u}\) from \(\mathbf{v}\):\[\mathbf{v} - 3\mathbf{u} = (-3, 2) - (-6, -3) = (-3 + 6, 2 + 3) = (3, 5)\]

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in vector math. In simple terms, it involves taking a vector and a scalar, which is a single number, and multiplying every component of the vector by that scalar.
For example, consider the vector \( \mathbf{u} = \langle -2, -1 \rangle \). If we want to find \( 2 \mathbf{u} \), we multiply each component of \( \mathbf{u} \) by 2. This gives us:

• Multiply \(-2\) (the first component of \(\mathbf{u}\)) by 2, resulting in \(-4\).
• Multiply \(-1\) (the second component of \(\mathbf{u}\)) by 2, resulting in \(-2\).

Therefore, \( 2 \mathbf{u} = \langle -4, -2 \rangle \).
This operation helps in scaling the vector. Scaling means making a vector longer or shorter, without changing its direction (if the scalar is positive) or reversing its direction (if the scalar is negative). This fundamental operation is used in many areas of mathematics and physics.

Vector Addition

Vector addition is another essential vector operation, one where you combine two vectors to create a new one. This is done by adding their corresponding components.
For instance, to find \( 2\mathbf{u} + 3\mathbf{v} \), we first determine what \( 3\mathbf{v} \) is by multiplying \( \mathbf{v} = \langle -3, 2 \rangle \) by 3:

• Multiply \(-3\) (first component of \(\mathbf{v}\)) by 3, yielding \(-9\).
• Multiply \(2\) (second component of \(\mathbf{v}\)) by 3, yielding \(6\).

This means \( 3\mathbf{v} = \langle -9, 6 \rangle \).
Now, to add \( 2\mathbf{u} = \langle -4, -2 \rangle \) and \( 3\mathbf{v} = \langle -9, 6 \rangle \):

• Add \(-4\) and \(-9\) to find the first component, obtaining \(-13\).
• Add \(-2\) and \(6\) to find the second component, obtaining \(4\).

Thus, \( 2\mathbf{u} + 3\mathbf{v} = \langle -13, 4 \rangle \).
This operation is very useful for determining the resultant vector when two forces or movements are applied in physical systems.

Vector Subtraction

Vector subtraction involves finding the difference between two vectors. It’s similar to vector addition, but instead of adding, you subtract the corresponding components.
To calculate \( \mathbf{v} - 3\mathbf{u} \), we first figure out \( 3\mathbf{u} \) by multiplying the components of \( \mathbf{u} = \langle -2, -1 \rangle \) by 3:

• Multiply \(-2\) by 3 to get \(-6\).
• Multiply \(-1\) by 3 to get \(-3\).

Hence, \( 3\mathbf{u} = \langle -6, -3 \rangle \).
Next, perform the vector subtraction by subtracting each component of \( 3\mathbf{u} \) from the corresponding component of \( \mathbf{v} \):

• Subtract \(-6\) from \(-3\) to get \(3\).
• Subtract \(-3\) from \(2\) to get \(5\).

Thus, the result is \( \mathbf{v} - 3\mathbf{u} = \langle 3, 5 \rangle \).
Vector subtraction is crucial in determining the net change when considering opposite directions or effects, such as in physics for analyzing forces or velocities.