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Chapter 11: Problem 60

Give geometric descriptions of the operations of addition of vectors and multiplication of a vector by a scalar.

Short Answer

Expert verified
Geometrically, the addition of two vectors results in a vector that is a diagonal of a parallelogram whose sides represent the two added vectors. Scalar multiplication of a vector changes its magnitude by the absolute value of the scalar, and changes its direction depending on whether the scalar is positive (same direction) or negative (opposite direction).

Step by step solution

01

Geometric Description of Vector Addition

Geometrically, the addition of two vectors creates a resultant vector, which is obtained by placing the initial point (or tail) of the second vector at the terminal point (or head) of the first vector, and then drawing a vector from the initial point of the first vector to the terminal point of the second vector. This can also be represented as a parallelogram: given vectors \( \mathbf{a} \) and \( \mathbf{b} \), the geometric representation of \( \mathbf{a} + \mathbf{b} \) is a diagonal of a parallelogram whose sides are \( \mathbf{a} \) and \( \mathbf{b} \). The direction and magnitude of the resultant vector depend on the directions and magnitudes of the two added vectors.
02

Geometric Description of Scalar Multiplication of a Vector

When a vector is multiplied by a scalar, the direction of the vector remains the same if the scalar is positive but the direction is reversed if the scalar is negative; the magnitude (or length) of the vector is multiplied by the absolute value of the scalar. For instance, if vector \( \mathbf{a} \) is multiplied by a scalar \( k \), then the resultant vector is \( k\mathbf{a} \). If \( k > 0 \), then the direction of \( k\mathbf{a} \) is the same as \( \mathbf{a} \), but the magnitude is \( k \) times that of \( \mathbf{a} \). If \( k < 0 \), then \( k\mathbf{a} \) has the opposite direction to \( \mathbf{a} \) and the magnitude is still \( k \) times that of \( \mathbf{a} \).

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