Problem 33 - Algebra and Trigonometry

Question

\(31-36\) Find \(2 \mathbf{u},-3 \mathbf{v}, \mathbf{u}+\mathbf{v},\) and \(3 \mathbf{u}-4 \mathbf{v}\) for the given vectors \(\mathbf{u}\) and \(\mathbf{v} .\) $$ \mathbf{u}=\langle 0,-1\rangle, \quad \mathbf{v}=\langle- 2,0\rangle $$

Step-by-Step Solution

Answer

\(2\mathbf{u} = \langle 0, -2 \rangle; -3\mathbf{v} = \langle 6, 0 \rangle; \mathbf{u} + \mathbf{v} = \langle -2, -1 \rangle; 3\mathbf{u} - 4\mathbf{v} = \langle 8, -3 \rangle\).

Step 1: Find \(2\mathbf{u}\)

To find \(2\mathbf{u}\), multiply each component of \(\mathbf{u}\) by 2. Given \(\mathbf{u} = \langle 0, -1 \rangle\), we calculate \(2 \cdot \langle 0, -1 \rangle = \langle 2 \cdot 0, 2 \cdot (-1) \rangle = \langle 0, -2 \rangle\).

Step 2: Find \(-3\mathbf{v}\)

To find \(-3\mathbf{v}\), multiply each component of \(\mathbf{v}\) by -3. Given \(\mathbf{v} = \langle -2, 0 \rangle\), we calculate \(-3 \cdot \langle -2, 0 \rangle = \langle -3 \cdot (-2), -3 \cdot 0 \rangle = \langle 6, 0 \rangle\).

Step 3: Find \(\mathbf{u} + \mathbf{v}\)

To find \(\mathbf{u} + \mathbf{v}\), add the corresponding components of \(\mathbf{u}\) and \(\mathbf{v}\). Given \(\mathbf{u} = \langle 0, -1 \rangle\) and \(\mathbf{v} = \langle -2, 0 \rangle\), we calculate \(\langle 0, -1 \rangle + \langle -2, 0 \rangle = \langle 0 + (-2), -1 + 0 \rangle = \langle -2, -1 \rangle\).

Step 4: Find \(3\mathbf{u} - 4\mathbf{v}\)

To find \(3\mathbf{u} - 4\mathbf{v}\), first multiply each component of \(\mathbf{u}\) by 3 and \(\mathbf{v}\) by 4, then subtract the result of \(4\mathbf{v}\) from \(3\mathbf{u}\). Given \(\mathbf{u} = \langle 0, -1 \rangle\) and \(\mathbf{v} = \langle -2, 0 \rangle\), calculate:- \(3\mathbf{u} = 3 \cdot \langle 0, -1 \rangle = \langle 0, -3 \rangle\)- \(4\mathbf{v} = 4 \cdot \langle -2, 0 \rangle = \langle -8, 0 \rangle\)Finally, subtract: \(\langle 0, -3 \rangle - \langle -8, 0 \rangle = \langle 0 - (-8), -3 - 0 \rangle = \langle 8, -3 \rangle\).

Key Concepts

Scalar Multiplication

Scalar multiplication is an essential operation in vector algebra. It involves multiplying a vector by a scalar (a real number), altering the magnitude of the vector without affecting its direction unless the scalar is negative. For example, if you have a vector \( \mathbf{u} = \langle 0, -1 \rangle \), multiplying it by a scalar like 2 involves performing the operation on each component individually:
\[ 2 \cdot \mathbf{u} = 2 \cdot \langle 0, -1 \rangle = \langle 2 \cdot 0, 2 \cdot (-1) \rangle = \langle 0, -2 \rangle \]
This operation changes the vector's length by the factor of the scalar, while maintaining its direction if positive. In essence, scalar multiplication stretches or shrinks the vector, and a negative scalar reverses its direction.
Here are some quick points to remember about scalar multiplication:

• The product of a vector and a scalar is a vector.
• If the scalar is positive, the direction remains the same.
• If the scalar is negative, the direction is inverted.

Understanding scalar multiplication is crucial as it's often used in more complex vector operations.

Vector Addition

Vector addition involves combining two vectors to produce a third vector. This operation is foundational in vector mathematics because it allows for the cumulative effects of different vector quantities. To add vectors \( \mathbf{u} = \langle 0, -1 \rangle \) and \( \mathbf{v} = \langle -2, 0 \rangle \), add their corresponding components:
\[ \mathbf{u} + \mathbf{v} = \langle 0 + (-2), -1 + 0 \rangle = \langle -2, -1 \rangle \]
Vector addition can be visualized by placing the tail of the second vector at the head of the first vector. The resultant vector is then drawn from the tail of the first to the head of the second.
Key aspects of vector addition include:

• Commutative Property: \( \mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u} \)
• Associative Property: \( (\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}) \)

This means that the order of addition does not affect the resultant vector, allowing flexibility in how vectors are grouped and combined in equations.

Subtraction of Vectors

Vector subtraction is similar to vector addition but involves finding the difference rather than the total. It is used to determine the relative change between two vector quantities. Subtracting one vector from another means adding the negative of the vector being subtracted. For the expression \( 3\mathbf{u} - 4\mathbf{v} \), begin by finding the scalar multiples of each vector:
\[ 3\mathbf{u} = \langle 0, -3 \rangle, \quad 4\mathbf{v} = \langle -8, 0 \rangle \]
Then, compute the result by performing vector addition between \( 3\mathbf{u} \) and the negative of \( 4\mathbf{v} \):
\[ 3\mathbf{u} - 4\mathbf{v} = \langle 0, -3 \rangle - \langle -8, 0 \rangle = \langle 0 - (-8), -3 - 0 \rangle = \langle 8, -3 \rangle \]
When subtracting vectors, flip the direction of the vector being subtracted and then proceed with an addition.
Important points about vector subtraction:

• Subtraction can be visualized by reversing the direction of the vector being subtracted and then performing vector addition.
• It’s equivalent to adding the opposite vector, making understanding the concept of vector inversion essential.

Subtraction helps in determinative contexts like determining displacement or relative velocities.