Let 𝐲=[ 3 1 ] and 𝐮=[ 8 6 ] . Compute the distance from 𝐲 to the line through 𝐮

step 1 Okay, this question wants us to find the distance between these two vectors. So to do this, we n

Let 𝐲=[ 3 1 ] and 𝐮=[ 8 6 ] . Compute the...

Let $\mathbf{y}=\left[\begin{array}{l}{3} \\ {1}\end{array}\right]$ and $\mathbf{u}=\left[\begin{array}{l}{8} \\ {6}\end{array}\right]$ . Compute the distance from $\mathbf{y}$ to the line through $\mathbf{u}$ and the origin.

Key Concepts

Dot Product

The dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. This operation is fundamentally used to measure the extent to which two vectors point in the same direction, and is critical in calculating projections as well as determining angles between vectors.

Orthogonal Decomposition

Orthogonal decomposition involves breaking a vector into two components: one parallel to a given vector (or subspace) and the other perpendicular to it. This procedure is crucial for finding the distance from a point to a line, as the desired distance is computed as the magnitude of the vector remainder after removing the parallel component.

Distance from a Point to a Line

In a geometric context, the distance from a point to a line is defined as the shortest length connecting the point to any point on the line. When the line passes through the origin and is determined by a vector direction, this distance can be found by calculating the magnitude of the difference between the point and its projection onto the line, effectively using orthogonal decomposition.

Vector Projection

Vector projection refers to the process of projecting one vector onto another vector. This is done by finding the component of the first vector in the direction of the second vector, typically using the formula proj_u(y) = (y · u)/(u · u) * u. This concept is essential when expressing one vector in terms of another direction, thereby decomposing it into parallel and perpendicular components.

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