In Exercises 9–16, let u = 2i - j, v = 3i + j, and w = i + 4j. Find each specified scalar. (4u) ⋅ v

Guida verificata passo dopo passo

Step 1: Understand the problem. We need to find the dot product of the vector 4u with vector v.

Step 2: Calculate 4u by multiplying each component of vector u by 4. Given u = 2i - j, then 4u = 4(2i - j) = 8i - 4j.

Step 3: Write down vector v, which is given as v = 3i + j.

Step 4: Use the dot product formula: If a = ai + bj and b = ci + dj, then a ⋅ b = ac + bd.

Step 5: Substitute the components of 4u and v into the dot product formula: (8i - 4j) ⋅ (3i + j) = (8)(3) + (-4)(1).

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Vector Operations

Understanding vector operations is crucial in this problem. Vectors can be added, subtracted, and multiplied by scalars. In this case, the vector u is being multiplied by the scalar 4, which scales the vector's magnitude while maintaining its direction. This operation is foundational for further calculations involving dot products.

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Algebraic Operations on Vectors

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The dot product is a key operation in vector algebra that combines two vectors to produce a scalar. It is calculated by multiplying corresponding components of the vectors and summing the results. The dot product provides insights into the angle between vectors and is essential for determining orthogonality and projection.

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Component Form of Vectors

Vectors are often expressed in component form, which involves breaking them down into their respective i (horizontal) and j (vertical) components. For example, the vector u = 2i - j has components 2 and -1. Understanding this representation is vital for performing operations like the dot product, as it allows for straightforward calculations using the individual components.

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