Problem 113 - Algebra and Trigonometry Real Mathematics, Real People

Question

Give geometric descriptions of (a) vector addition and (b) scalar multiplication.

Step-by-Step Solution

Answer

Geometrically, vector addition is the process of joining two vectors head-to-tail and drawing the resultant from the tail of the first to the head of the last vector. Scalar multiplication involves changing the magnitude of a vector by a scalar, keeping the direction the same if scalar is positive, and reversing if negative.

Step 1: Geometric Description of Vector Addition

Vector addition geometrically occurs when two vectors are placed head-to-tail and a resultant vector is drawn from the tail of the first vector to the head of the second vector. This resultant vector represents the sum of the two vectors. The addition of vectors \(\vec{v}\) and \(\vec{u}\) can be represented graphically by drawing \(\vec{v}\) and \(\vec{u}\) such that the head of \(\vec{v}\) is at the tail of \(\vec{u}\). The resultant vector, denoted \(\vec{v} + \vec{u}\), originates from the tail of \(\vec{v}\) and ends at the head of \(\vec{u}\).

Step 2: Geometric Description of Scalar Multiplication

Scalar multiplication of a vector \(\vec{v}\) by a scalar k involves changing the magnitude of the vector by a factor of k, but not the direction if k is positive. When k is negative, the direction is reversed. If \(\vec{v}\) is the original vector, the vector k\(\vec{v}\) is the vector that has k times the length of \(\vec{v}\) but in the same (or opposite if k is negative) direction.