In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar.
||-2v||

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Identify the given vector \( \mathbf{v} = \mathbf{i} - 5\mathbf{j} \), which can be written in component form as \( \mathbf{v} = (1, -5) \).

Calculate the scalar multiplication of the vector \( \mathbf{v} \) by \( -2 \), which gives \( -2\mathbf{v} = -2(1, -5) = (-2, 10) \).

Recall that the magnitude (or norm) of a vector \( \mathbf{a} = (x, y) \) is given by the formula: \[ \\| \mathbf{a} \\| = \sqrt{x^2 + y^2} \]

Apply the magnitude formula to the vector \( -2\mathbf{v} = (-2, 10) \) by substituting \( x = -2 \) and \( y = 10 \) into the formula: \[ \\| -2\mathbf{v} \\| = \sqrt{(-2)^2 + 10^2} \]

Simplify the expression under the square root to find the magnitude of \( -2\mathbf{v} \).

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Vector Notation and Components

Vectors are represented using unit vectors i and j to denote their components along the x and y axes, respectively. For example, v = i - 5j means the vector has an x-component of 1 and a y-component of -5. Understanding this notation is essential for performing vector operations.

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying each component of a vector by a scalar value. For instance, multiplying vector v by -2 results in a new vector with components scaled by -2, changing both magnitude and direction accordingly.

Magnitude (Norm) of a Vector

The magnitude of a vector is the length or size of the vector, calculated using the Pythagorean theorem as the square root of the sum of the squares of its components. For vector v = (x, y), ||v|| = √(x² + y²). This concept is crucial for finding ||-2v||.

In Exercises 27–30, let v = i - 5j and w = -2i + 7j. Find each specified vector or scalar.||-2v||