Question

In each case either prove the statement or give an example showing that it is false. a. The zero vector \(\mathbf{0}\) is the only vector of length 0 . b. If \(\|\mathbf{v}-\mathbf{w}\|=0,\) then \(\mathbf{v}=\mathbf{w}\). c. If \(\mathbf{v}=-\mathbf{v},\) then \(\mathbf{v}=\mathbf{0}\). d. If \(\|\mathbf{v}\|=\|\mathbf{w}\|,\) then \(\mathbf{v}=\mathbf{w}\). e. If \(\|\mathbf{v}\|=\|\mathbf{w}\|,\) then \(\mathbf{v}=\pm \mathbf{w}\). f. If \(\mathbf{v}=t \mathbf{w}\) for some scalar \(t,\) then \(\mathbf{v}\) and \(\mathbf{w}\) have the same direction. \(\mathrm{g}\). If \(\mathbf{v}, \mathbf{w},\) and \(\mathbf{v}+\mathbf{w}\) are nonzero, and \(\mathbf{v}\) and \(\mathbf{v}+\mathbf{w}\) parallel, then \(\mathbf{v}\) and \(\mathbf{w}\) are parallel. h. \(\|-5 \mathbf{v}\|=-5\|\mathbf{v}\|,\) for all \(\mathbf{v}\). i. If \(\|\mathbf{v}\|=\|2 \mathbf{v}\|,\) then \(\mathbf{v}=\mathbf{0}\). j. \(\|\mathbf{v}+\mathbf{w}\|=\|\mathbf{v}\|+\|\mathbf{w}\|,\) for all \(\mathbf{v}\) and \(\mathbf{w}\).

Short answer

a: true, b: true, c: true, d: false, e: true, f: true, g: true, h: false, i: true, j: false.

Step-by-step solution

Analyzing statement a

We know that the length or magnitude of a vector \( \mathbf{v} \) is given by \( \|\mathbf{v}\| \), which is always non-negative. By definition, the length is zero only if the vector itself is zero, \( \mathbf{v} = \mathbf{0} \). Thus, the zero vector \( \mathbf{0} \) is indeed the only vector of length 0. This statement is true.

Analyzing statement b

The magnitude of the difference between two vectors \( \mathbf{v} \) and \( \mathbf{w} \) is 0, i.e., \( \|\mathbf{v} - \mathbf{w}\| = 0 \), only if \( \mathbf{v} - \mathbf{w} = \mathbf{0} \) which means \( \mathbf{v} = \mathbf{w} \). Therefore, this statement is true.

Analyzing statement c

If \( \mathbf{v} = -\mathbf{v} \), adding \( \mathbf{v} \) to both sides gives \( \mathbf{v} + \mathbf{v} = \mathbf{0} \), which simplifies to \( 2\mathbf{v} = \mathbf{0} \). This implies \( \mathbf{v} = \mathbf{0} \). Therefore, this statement is true.

Analyzing statement d

Knowing \( \|\mathbf{v}\| = \|\mathbf{w}\| \) does not imply \( \mathbf{v} = \mathbf{w} \). For example, if \( \mathbf{v} = (1, 0) \) and \( \mathbf{w} = (0, 1) \), both have the same magnitude of 1 but are not equal. Therefore, this statement is false.

Analyzing statement e

Given \( \|\mathbf{v}\| = \|\mathbf{w}\| \), \( \mathbf{v} \) can be \( \mathbf{w} \) or \(-\mathbf{w}\) as they have the same magnitude, like \( \mathbf{v} = (1, 0) \) and \( \mathbf{w} = (1, 0) \) or \( \mathbf{w} = (-1, 0) \). Hence, \( \mathbf{v} = \pm \mathbf{w} \). This statement is true.

Analyzing statement f

If \( \mathbf{v} = t\mathbf{w} \) for some scalar \( t \), then \( \mathbf{v} \) and \( \mathbf{w} \) are scalar multiples of each other, hence they have the same or opposite direction depending on the sign of \( t \). This statement is true.

Analyzing statement g

If \( \mathbf{v} \) is parallel to \( \mathbf{v} + \mathbf{w} \), then \( \mathbf{v} + \mathbf{w} = k\mathbf{v} \) for some scalar \( k \). Rearranging gives \( \mathbf{w} = (k-1)\mathbf{v} \), thus \( \mathbf{w} \) is a scalar multiple of \( \mathbf{v} \) and hence they are parallel. This statement is true.

Analyzing statement h

\( \|-5\mathbf{v}\| = 5\|\mathbf{v}\| \), since the magnitude of a vector scaled by a negative scalar is the same as when scaled by its positive. \(-5\|\mathbf{v}\|\) gives a negative value, which isn't correct for magnitudes. Hence, \|-5\mathbf{v}\| = 5\|\mathbf{v}\|. This statement is false.

Analyzing statement i

If \( \|\mathbf{v}\| = \|2 \mathbf{v}\| \), then \( \|2 \mathbf{v}\| = 2\|\mathbf{v}\| \). Equating gives that \( 2\|\mathbf{v}\| = \|\mathbf{v}\| \), implying \( \|\mathbf{v}\| = 0 \), therefore \( \mathbf{v} = \mathbf{0} \). This statement is true.

Analyzing statement j

The triangle inequality states \( \|\mathbf{v} + \mathbf{w}\| \leq \|\mathbf{v}\| + \|\mathbf{w}\| \). The equality holds only if \( \mathbf{v} \) and \( \mathbf{w} \) are in the same or opposite direction. For all \( \mathbf{v} \) and \( \mathbf{w} \), the statement is false.

Key concepts

Vector Magnitude

The magnitude of a vector, often termed as its length, is a measure of how 'long' the vector is. For a vector \( \mathbf{v} = (v_1, v_2, \ldots, v_n) \), its magnitude is calculated using the following formula: \[ \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2} \]

• The magnitude is always a non-negative value.
• A vector is considered to have zero magnitude if and only if it is the zero vector \( \mathbf{0} = (0, 0, \ldots, 0) \).
• The magnitude signifies distance in geometric space, akin to how you measure length along a ruler.

Understanding vector magnitude is crucial as it helps in determining distance, normalizing vectors, and assessing vector direction.

Scalar Multiplication

Scalar multiplication is a process of scaling a vector by multiplying it with a scalar (a number). If you have a vector \( \mathbf{v} = (v_1, v_2, \ldots, v_n) \) and a scalar \( t \), the result of multiplying the vector by the scalar is: \[ t \mathbf{v} = (tv_1, tv_2, \ldots, tv_n) \]

• This operation stretches or shrinks the vector in the direction of the original vector based on the absolute value of the scalar.
• If \( t = 0 \), the product is the zero vector.
• If \( t = 1 \), the vector remains unchanged.
• Both the direction and magnitude may change, but the line of the vector always passes through the origin.

Scalar multiplication is essential for tasks such as adjusting forces in physics or scaling transformations in graphics.

Triangle Inequality

The triangle inequality is a fundamental theorem in vector algebra. For two vectors \( \mathbf{v} \) and \( \mathbf{w} \), the triangle inequality states: \[ \| \mathbf{v} + \mathbf{w} \| \leq \| \mathbf{v} \| + \| \mathbf{w} \| \]

• This means that the length (magnitude) of the sum of two vectors is never greater than the sum of their individual lengths.
• The equality holds, meaning \( \| \mathbf{v} + \mathbf{w} \| = \| \mathbf{v} \| + \| \mathbf{w} \| \), if and only if the vectors are collinear (parallel, meaning they lie on the same line in space).
• The principle is analogous to the fact that, in a triangle, the length of any two sides added together is always greater than or equal to the length of the third side.

This inequality is useful in many mathematical proofs and is widely applied in fields such as physics, engineering, and computer science.

Parallel Vectors

Vectors are parallel if they have the same or exactly opposite direction. For vectors \( \mathbf{v} \) and \( \mathbf{w} \), they are considered parallel if there is some scalar \( t \) such that: \[ \mathbf{v} = t \mathbf{w} \]

• If \( t > 0 \), vectors have the same direction.
• If \( t < 0 \), they point in opposite directions.
• If vectors are parallel and not equal to zero, they lie on the same line or plane in geometric space.
• Parallel vectors share the same or reverse orientation without necessarily being identical.

Understanding parallel vectors is crucial for solving problems related to angles, forces, and directional movement in physics and engineering.