Problem 13 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the vectors \(\mathbf{u}=\langle 2,2\rangle, \mathbf{v}=\langle-5,3\rangle,\) and \(\mathbf{w}=\langle\mathbf{1},-\mathbf{4}\rangle\) to find the indicated quantity. State whether the result is a vector or a scalar. $$\mathbf{u} \cdot 2 \mathbf{v}$$

Step-by-Step Solution

Answer

-8. The dot product result is a scalar not a vector.

Step 1: Scalar Multiply \(\mathbf{v}\)

Multiply vector \(\mathbf{v}\)=\(\langle-5,3\rangle\) by scalar 2 to yield a new vector, \(2\mathbf{v}\)=\(\langle-10,6\rangle\) .

Step 2: Calculate Dot Product

Take the dot product of vectors \(\mathbf{u}\)=\(\langle 2,2\rangle\) and \(2\mathbf{v}\)=\(\langle-10,6\rangle\), which is \(\mathbf{u} \cdot 2 \mathbf{v} = u_1*v_1 + u_2*v_2 = 2*(-10) + 2*6\).

Key Concepts

Dot Product

The dot product is a fundamental operation you can perform on two vectors. It gives a scalar result rather than a vector, representing how much one vector extends in the direction of another. Given vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), the dot product is calculated as:\[ \mathbf{a} \cdot \mathbf{b} = a_1 \times b_1 + a_2 \times b_2 \]

• This is why the result is called a "scalar" because it reduces the dimensional space of vectors to a single number, or magnitude.
• The dot product can tell you if two vectors are perpendicular. If the dot product is zero, the vectors are orthogonal.

In our exercise example, taking \( \mathbf{u} = \langle 2,2 \rangle \) and \( 2\mathbf{v} = \langle -10,6 \rangle \), the result is:

• 2 \( \times \) -10 = -20
• 2 \( \times \) 6 = 12

Adding these gives \(-20 + 12 = -8\), which is a scalar.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a constant number), affecting the vector’s magnitude while keeping its direction unchanged. When a vector \( \mathbf{v} = \langle v_1, v_2 \rangle \) is multiplied by scalar \( c \), the resulting vector is:\[ c\mathbf{v} = \langle c \times v_1, c \times v_2 \rangle \]

• If \( c \) is positive, the direction remains the same.
• If \( c \) is negative, the direction is inverted.
• Zero scalar results in a zero vector.

In our problem, the vector \( \mathbf{v} = \langle -5,3 \rangle \) becomes \( 2\mathbf{v} = \langle -10,6 \rangle \) when each component is multiplied by 2. This does not change its direction, just elongates it by a factor of 2.

Vectors in Mathematics

Vectors are quantities that have both magnitude and direction, and they are crucial in multiple fields including physics and engineering. Represented usually by an arrow in geometry, vectors can be defined in two-dimensional or three-dimensional space.

• A vector is typically denoted as \( \mathbf{v} = \langle v_1, v_2 \rangle \) in 2D, containing x and y components.
• They are used to represent physical quantities like force, velocity, or displacement where direction matters.

Understanding vector operations like addition, subtraction, scalar multiplication, and dot products is vital since they allow manipulation of vectors in mathematical expressions and problem-solving. Each operation follows specific rules that transform vectors within their dimensional space, enabling complex simulations and analysis. Being comfortable with these foundational concepts smoothly aids in more advanced vector dynamics and mathematical computations.