91Ó°ÊÓ

Vector addition is a fundamental operation where two or more vectors are combined. Unlike scalar quantities that simply add up, vectors follow certain rules. When adding vectors, we add their respective components. Imagine you have a vector \( \mathbf{a} = \langle -1, -2, 4 \rangle \) and another vector \( \mathbf{b} = \langle -5, 6, -7 \rangle \). To find \( \mathbf{a} + \mathbf{b} \), add each corresponding component separately:

• First component: \( -1 + (-5) = -6 \)
• Second component: \( -2 + 6 = 4 \)
• Third component: \( 4 + (-7) = -3 \)

Thus, \( \mathbf{a} + \mathbf{b} = \langle -6, 4, -3 \rangle \). It's like performing simple arithmetic operations within each dimension. By following this method, you will be able to find the resultant vector. Vector addition can be visualized as placing the tail of one vector at the head of the other and drawing a new vector from the tail of the first to the head of the second.

Scalar multiplication involves multiplying a vector by a number (a scalar). This operation stretches or shrinks the vector without changing its direction (unless the scalar is negative, which also reverses the direction). Let’s find \( 4\mathbf{a} \) for the vector \( \mathbf{a} = \langle -1, -2, 4 \rangle \). Multiply each component of the vector by 4:

• \( 4 \times (-1) = -4 \)
• \( 4 \times (-2) = -8 \)
• \( 4 \times 4 = 16 \)

Thus, we get \( 4\mathbf{a} = \langle -4, -8, 16 \rangle \).
This shows each element of the vector scaled by 4. Scalar multiplication alters the magnitude of the vector, not the direction, and helps in general transformations where scaling is necessary.

The component form is a way of expressing a vector by its components. This breaks down a vector into its essential parts, making operations like addition and scaling straightforward. For example, if a vector \( \mathbf{a} \) is expressed as \( \langle -1, -2, 4 \rangle \), each number inside the angle brackets represents a component of the vector aligned with the axes of a coordinate system.To find \( -5\mathbf{a} + 3\mathbf{b} \), we first find \( -5\mathbf{a} \) by multiplying each component of \( \mathbf{a} \) by -5:

• \( -5 \times (-1) = 5 \)
• \( -5 \times (-2) = 10 \)
• \( -5 \times 4 = -20 \)

Resulting in \( -5\mathbf{a} = \langle 5, 10, -20 \rangle \).Next, for \( 3\mathbf{b} \), multiply each component of \( \mathbf{b} = \langle -5, 6, -7 \rangle \) by 3:

• \( 3 \times (-5) = -15 \)
• \( 3 \times 6 = 18 \)
• \( 3 \times (-7) = -21 \)

So, \( 3\mathbf{b} = \langle -15, 18, -21 \rangle \).Finally, add them for \( -5\mathbf{a} + 3\mathbf{b} \):

• \( 5 + (-15) = -10 \)
• \( 10 + 18 = 28 \)
• \( -20 + (-21) = -41 \)

Giving \( -5\mathbf{a} + 3\mathbf{b} = \langle -10, 28, -41 \rangle \). The component form helps visualize each individual influence along various axes, making it a powerful tool in vector mathematics.