Problem 47 - Algebra and Trigonometry

Question

Find each quantity if \(\mathbf{v}=3 \mathbf{i}-5 \mathbf{j}\) and \(\mathbf{w}=-2 \mathbf{i}+3 \mathbf{j}\) \(\|\mathbf{v}\|-\|\mathbf{w}\|\)

Step-by-Step Solution

Answer

\( \sqrt{34} - \sqrt{13} \)

Step 1 - Determine the Magnitude of \(\textbf{v}\)

Given \(\textbf{v}=3 \mathbf{i}-5 \mathbf{j}\), use the formula for the magnitude of a vector: \[ \|\textbf{v}\| = \sqrt{3^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34}. \] So, \(\textbf{v}\| = \sqrt{34}\).

Step 2 - Determine the Magnitude of \(\textbf{w}\)

Given \(\textbf{w}=-2 \mathbf{i}+3 \mathbf{j}\), use the same formula for the magnitude: \[ \|\textbf{w}\| = \sqrt{(-2)^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}. \] So, \(\textbf{w}\| = \sqrt{13}\).

Step 3 - Calculate the Difference of the Magnitudes

Now find the difference between the magnitudes of \(\textbf{v}\) and \(\textbf{w}\): \[ \|\textbf{v}\| - \|\textbf{w}\| = \sqrt{34} - \sqrt{13}. \]

Key Concepts

magnitude formula

In mathematics, the magnitude of a vector represents its length or size. It is a crucial concept for understanding vectors in geometry and physics. To find the magnitude of a vector, you use the Pythagorean theorem. The magnitude formula is given as follows: \ \ \[ \| \mathbf{v} \| = \sqrt{v_1^2 + v_2^2} \] Here, \ \ \(v_1\) and \(v_2\) are the components of the vector \(\mathbf{v}\) in the i and j directions respectively. For example, for the vector \(\mathbf{v} = 3\mathbf{i} - 5\mathbf{j}\), you calculate its magnitude as follows: \ \ \[\mathbf{v} = \sqrt{3^2 + (-5)^2} = \sqrt{9 + 25} = \sqrt{34}\] Similarly, for \(\mathbf{w} = -2\mathbf{i} + 3 \mathbf{j}\), the magnitude would be: \[ \| \mathbf{w} \| = \sqrt{(-2)^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \] Calculating the magnitude is often the first step in many vector problems.

vector subtraction

Vector subtraction involves finding the difference between two vectors. This operation is important in various applications, such as determining relative positions and displacements. When subtracting vectors, you subtract their corresponding components: \ \ \[\mathbf{v} - \mathbf{w} = (v_1 - w_1) \mathbf{i} + (v_2 - w_2) \mathbf{j} \] For the vectors given, \(\mathbf{v} = 3 \mathbf{i} - 5 \mathbf{j} \) and \( \mathbf{w} = -2 \mathbf{i} + 3 \mathbf{j} \): \ \ The subtraction would be: \ \ \[ (3 - (-2)) \mathbf{i} + (-5 - 3) \mathbf{j} = 5 \mathbf{i} - 8 \mathbf{j} \] This operation is straightforward but fundamental to understanding vector dynamics in physics and engineering.

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle. It's not always about triangles though; this theorem also underpins the calculation of vector magnitudes. The Pythagorean theorem states: \ \ \[a^2 + b^2 = c^2\] Here, \(a\) and \(b\) are the lengths of the triangle's legs, and \(c\) is the hypotenuse. When applied to vectors, \(a\) and \(b\) are the components of the vector, and \(c\) represents the magnitude of the vector. For instance, for any vector \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} \), the magnitude \(\|\mathbf{v}\|\) is calculated using: \ \ \[ \| \mathbf{v} \| = \sqrt{v_1^2 + v_2^2} \] Thus, finding the magnitude of vectors \( \mathbf{v} \) and \( \mathbf{w} \) translates to finding the hypotenuse of a right-angled triangle formed by the vector's components. This way, the Pythagorean theorem becomes an indispensable tool in vector mathematics.