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Ch 01: Units, Physical Quantities & Vectors
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 14th EditionCh 01: Units, Physical Quantities & VectorsProblema 38a
Capitolo 1, Problema 38a

Given two vectors A = 4i + 7j and B = 5i - 2j, find the magnitude of each vector.

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To find the magnitude of a vector, use the formula: \( |\mathbf{A}| = \sqrt{A_x^2 + A_y^2} \), where \( A_x \) and \( A_y \) are the components of vector \( \mathbf{A} \).
For vector \( \mathbf{A} = 4\mathbf{i} + 7\mathbf{j} \), identify the components: \( A_x = 4 \) and \( A_y = 7 \).
Substitute the components of vector \( \mathbf{A} \) into the magnitude formula: \( |\mathbf{A}| = \sqrt{4^2 + 7^2} \).
Similarly, for vector \( \mathbf{B} = 5\mathbf{i} - 2\mathbf{j} \), identify the components: \( B_x = 5 \) and \( B_y = -2 \).
Substitute the components of vector \( \mathbf{B} \) into the magnitude formula: \( |\mathbf{B}| = \sqrt{5^2 + (-2)^2} \).

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Vector Magnitude

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem. For a vector A = ai + bj, the magnitude is given by |A| = √(a² + b²). This formula helps determine the size of the vector in the coordinate plane.
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Calculating Magnitude & Components of a Vector

Vector Components

Vectors are represented by components along the coordinate axes, typically denoted as i and j for the x and y axes, respectively. Understanding vector components is crucial for calculating vector magnitude and performing operations like addition or subtraction.
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Vector Addition By Components

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle. It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides, which is essential for calculating vector magnitudes.
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Parallel Axis Theorem
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