Ch 03: Vectors and Coordinate Systems

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo

Knight Calc 5th Edition

Ch 03: Vectors and Coordinate Systems

Problema 17c

Capitolo 3, Problema 17c

Let E = 2i + 3j and F = 2i - 2j. Find the magnitude of -E - 2F.

Guida verificata passo dopo passo

Step 1: Write down the given vectors. E = 2i + 3j and F = 2i - 2j.

Step 2: Compute the expression -E. To negate a vector, multiply each component by -1. This gives -E = -2i - 3j.

Step 3: Compute the expression -2F. Multiply each component of F by -2. This gives -2F = -4i + 4j.

Step 4: Add the vectors -E and -2F. Combine the i-components and j-components separately: (-2i - 3j) + (-4i + 4j) = (-2 - 4)i + (-3 + 4)j = -6i + 1j.

Step 5: Find the magnitude of the resulting vector -6i + 1j. Use the formula for the magnitude of a vector: |V| = sqrt((x-component)^2 + (y-component)^2). Substitute the components: |V| = sqrt((-6)^2 + (1)^2).

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Vector Addition and Subtraction

Vector addition and subtraction involve combining vectors by adding or subtracting their corresponding components. For example, if vector E has components (2, 3) and vector F has components (2, -2), then -E would be (-2, -3) and -2F would be (-4, 4). The resultant vector is found by summing these component-wise.

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Subtracting Vectors Graphically

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the formula √(x² + y²) for a 2D vector with components (x, y). This value represents the distance from the origin to the point defined by the vector in the Cartesian plane. For example, the magnitude of vector (a, b) is √(a² + b²).

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Calculating Magnitude & Components of a Vector

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying a vector by a scalar (a real number), which scales the vector's magnitude without changing its direction. For instance, multiplying vector F by -2 scales its components by -2, effectively reversing its direction and doubling its length. This operation is crucial for manipulating vectors in various physics problems.

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Introduction to Vectors and Scalars

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