91Ó°ÊÓ

In mathematics, vector addition is a fundamental operation used to combine two vectors to produce a third. In the simplest version, if you have vectors \( \mathbf{a} \) and \( \mathbf{b} \), vector addition involves adding each corresponding component of these vectors.
Consider the example from the exercise: - \( \mathbf{a} = \langle 3, -2, 4 \rangle \) - \( \mathbf{b} = \langle -5, 6, -9 \rangle \) The addition process is quite straightforward. You take the first components of each vector—3 from \( \mathbf{a} \) and -5 from \( \mathbf{b} \)—and add them together to get -2. You repeat this process for the second and third components.
So,

• First components: \( 3 + (-5) = -2 \)
• Second components: \( -2 + 6 = 4 \)
• Third components: \( 4 + (-9) = -5 \)

This gives the resulting vector: \( \langle -2, 4, -5 \rangle \).
Remember, vector addition is both commutative and associative, meaning you can add vectors in any order and group them as you see fit, and the result will be the same.

Scalar multiplication involves multiplying a vector by a scalar (a simple number), which stretches or shrinks the vector. Consider vector \( \mathbf{a} = \langle 3, -2, 4 \rangle \) from the exercise. If we multiply this vector by a scalar value, say 4, the operation is done by multiplying each component of \( \mathbf{a} \) by 4.
Here's how it happens step by step:

• The first component is \( 3 \times 4 = 12 \)
• The second component is \( -2 \times 4 = -8 \)
• The third component is \( 4 \times 4 = 16 \)

So, \( 4 \mathbf{a} \) becomes \( \langle 12, -8, 16 \rangle \).
It's important to understand that scalar multiplication changes the magnitude of the vector but not its direction (unless the scalar is negative, which reverses the direction). This operation is fundamental when dealing with vector spaces, as it allows us to scale vectors to any desired length while maintaining their direction.

Linear combinations involve creating a new vector from multiples of two or more vectors. For example, find the linear combination \(-5\mathbf{a} + 3\mathbf{b}\) by scaling each vector accordingly and then adding them.
Start with scaling:
- Vector \(-5 \mathbf{a}\):

• Multiply each component of \( \mathbf{a} = \langle 3, -2, 4 \rangle \) by \(-5\):
• \(-5\times3 = -15\)
• \(-5\times(-2) = 10\)
• \(-5\times4 = -20\)

This results in \( \langle -15, 10, -20 \rangle \)
Then, for vector \(3\mathbf{b}\):

• Multiply each component of \( \mathbf{b} = \langle -5, 6, -9 \rangle \) by 3:
• \(3\times(-5) = -15\)
• \(3\times6 = 18\)
• \(3\times(-9) = -27\)

This results in \( \langle -15, 18, -27 \rangle \)
Finally, add the resulting vectors:

• Components: \(-15 + (-15) = -30\)
• Components: \(10 + 18 = 28\)
• Components: \(-20 + (-27) = -47\)

Thus, the linear combination \(-5\mathbf{a} + 3\mathbf{b}\) results in \( \langle -30, 28, -47 \rangle \).
Linear combinations are crucial in understanding vector spaces, as they allow us to represent any vector in the space as a combination of a few basis vectors. This concept is widely used in various fields including physics, engineering, and computer graphics.