Let u = 〈-2, 1〉, v = 〈3, 4〉, and w = 〈-5, 12〉. Evaluate each expression.
(3u) • v

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Recall that the dot product of two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) is given by the formula: \[ \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \]

First, calculate the vector \( 3\mathbf{u} \) by multiplying each component of \( \mathbf{u} = \langle -2, 1 \rangle \) by 3: \[ 3\mathbf{u} = \langle 3 \times (-2), 3 \times 1 \rangle = \langle -6, 3 \rangle \]

Now, use the dot product formula to find \( (3\mathbf{u}) \cdot \mathbf{v} \), where \( \mathbf{v} = \langle 3, 4 \rangle \): \[ (3\mathbf{u}) \cdot \mathbf{v} = (-6)(3) + (3)(4) \]

Simplify the expression by performing the multiplications inside the dot product: \[ (-6)(3) + (3)(4) = -18 + 12 \]

Finally, add the results to get the value of the dot product: \[ -18 + 12 \]

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Vector Scalar Multiplication

Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). For example, multiplying vector u = 〈-2, 1〉 by 3 results in 〈-6, 3〉. This operation scales the vector's magnitude without changing its direction.

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Multiplying Vectors By Scalars

Dot Product of Vectors

The dot product of two vectors is the sum of the products of their corresponding components. For vectors a = 〈a1, a2〉 and b = 〈b1, b2〉, the dot product is a1*b1 + a2*b2. It results in a scalar and measures the extent to which the vectors point in the same direction.

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Introduction to Dot Product

Properties of Dot Product with Scalar Multiplication

The dot product is distributive over scalar multiplication, meaning (c*u) • v = c*(u • v), where c is a scalar. This property allows simplification by factoring out scalars before computing the dot product, making calculations more efficient.

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Pratica correlata

Domanda del libro di testo

Find the area of each triangle ABC.

a = 76.3 ft, b = 109 ft, c = 98.8 ft

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Domanda del libro di testo

Find the angle between each pair of vectors. Round to two decimal places as necessary.