Problem 85 - Algebra and Trigonometry Real Mathematics, Real People

Question

What can be said about the vectors \(\mathbf{u}\) and \(\mathbf{v}\) under each condition? (a) The projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(\mathbf{u}.\) (b) The projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(0 .\)

Step-by-Step Solution

Answer

In the first case (a), vector \(\mathbf{u}\) is collinear with vector \(\mathbf{v}\). In the second case (b), vector \(\mathbf{u}\) is orthogonal or perpendicular to vector \(\mathbf{v}\).

Step 1: Determine \(\mathbf{u}\) when its projection onto \(\mathbf{v}\) equals \(\mathbf{u}\)

If the projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(\mathbf{u}\), that means \(\mathbf{u}\) lies entirely in the direction of \(\mathbf{v}\). Essentially, \(\mathbf{u}\) and \(\mathbf{v}\) are scalar multiples of each other, which means they are in the same line or they are collinear.

Step 2: Determine \(\mathbf{u}\) when its projection onto \(\mathbf{v}\) equals \(0\)

If the projection of \(\mathbf{u}\) onto \(\mathbf{v}\) equals \(0\), then \(\mathbf{u}\) is perpendicular to \(\mathbf{v}\). This is because the projection of one vector onto another is zero only if the vectors are orthogonal.

Key Concepts

Vector Algebra

Vector algebra is a fundamental aspect of mathematics, particularly in physics and engineering, where it enables the representation and manipulation of quantities that have both magnitude and direction, such as force or velocity. A vector is visually represented as an arrow pointing from one point to another, with the length of the arrow indicating the vector’s magnitude and the direction of the arrow indicating its direction.

Collinear Vectors

Collinear vectors are vectors that lie on the same straight line, even if they are in opposite directions. In the context of vector projection, if the projection of vector \(\mathbf{u}\) onto vector \(\mathbf{v}\) is equal to \(\mathbf{u}\) itself, \(\mathbf{u}\) and \(\mathbf{v}\) are considered collinear. This implies that \(\mathbf{u}\) is simply a scalar multiple of \(\mathbf{v}\), which means we could write \(\mathbf{u} = k\cdot\mathbf{v}\), where \(k\) is a scalar value. Recognizing collinear vectors is essential in various applications, including determining lines of force in physics or aligning components in engineering designs.

Orthogonal Vectors

Orthogonal vectors, also known as perpendicular vectors, are vectors that meet at a right angle (90 degrees). In vector algebra, the orthogonality of vectors \(\mathbf{u}\) and \(\mathbf{v}\) signifies that they have no component of one in the direction of the other. If the projection of vector \(\mathbf{u}\) onto vector \(\mathbf{v}\) equals \(0\), it indicates that \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal. This means their dot product \(\mathbf{u}\cdot\mathbf{v}\) is zero. Orthogonal vectors are crucial in constructing coordinate systems, designing mechanisms that require right angles, and in simplifying complex vector calculations by decomposing vectors into mutually orthogonal components.