91Ó°ÊÓ

Scalar multiplication is a process where a vector is multiplied by a scalar, or a single number, affecting the vector's magnitude without changing its direction. This concept is critical when you're dealing with expressions like \( \lambda(\mathbf{u} + 2\mathbf{v}) \). Here, \( \lambda \) is the scalar, indicating how much to scale the vector \( \mathbf{u} + 2\mathbf{v} \).

• If \( \lambda \) is greater than 1, the vector will stretch.
• If \( \lambda \) is between 0 and 1, the vector will shrink.
• If \( \lambda \) is 0, the vector disappears, essentially becoming a zero vector.
• If \( \lambda \) is negative, the vector will not only change in magnitude but also reverse its direction.

For example, in the step-by-step solution, \( \lambda(\mathbf{u} + 2\mathbf{v}) \) becomes \( \lambda \mathbf{u} + 2\lambda \mathbf{v} \) through scalar multiplication, indicating the transformation effects on each component of the vector separately.

Understanding vector components is crucial to breaking down and analyzing vectors in different directions. Vectors can be decomposed into components, typically aligned along the main axes like \( x \) and \( y \) axes in a two-dimensional plane, or simply in terms of two non-equivalent vectors, such as \( \mathbf{u} \) and \( \mathbf{v} \) here.
When you have a vector like \( \mathbf{a} = 4\mathbf{u} + 5\mathbf{v} \), the vector components \( 4\mathbf{u} \) and \( 5\mathbf{v} \) indicate how much of each direction (\( \mathbf{u} \) and \( \mathbf{v} \)) contribute to the vector \( \mathbf{a} \). This shows us:

• 4 units in the direction of \( \mathbf{u} \)
• 5 units in the direction of \( \mathbf{v} \)

In this way, decomposing vectors into components allows you to simplify and resolve expressions like the ones found in your exercises, such as adding \( \lambda\mathbf{b} = \lambda\mathbf{u} + 2\lambda\mathbf{v} \) to other vectors.

Vector equivalence indicates that two vectors are equal when they have the same magnitude and direction. This does not mean that they must originate from the same point; rather, they must have the same size and point in the same direction. In the given exercise, equivalence tells us that the vectors \( \mathbf{a} + \lambda \mathbf{b} \) and \( \mathbf{u} - \mathbf{v} \) should, after scalar adjustments, travel the same path and point the same way.
To establish equivalence, each component of one vector needs to match the corresponding component of the other vector. In this exercise:

• For the component along \( \mathbf{u} \): Make sure \( 4 + \lambda = 1 \).
• For the component along \( \mathbf{v} \): Ensure \( 5 + 2\lambda = -1 \).

Once you've solved these equations for \( \lambda \), you determine the scalar needed to adjust \( \mathbf{b} \), ensuring \( \mathbf{a} + \lambda \mathbf{b} \) matches \( \mathbf{u} - \mathbf{v} \). This condition checks that both vectors will have matched both magnitude and direction, confirming their equivalence.