Problem 51 - Trigonometry

Question

If \(\mathbf{U}=7 \mathbf{i}+2 \mathbf{j}\) and \(\mathbf{V}=3 \mathbf{i}-4 \mathrm{j}\), find \(\mathbf{U} \cdot \mathbf{V}\). a. \(21 \mathbf{i}-8 \mathbf{j}\) b. \(21 \mathrm{i}^{2}-22 \mathrm{ij}-8 \mathrm{j}^{2}\) c. 2 d. 13

Step-by-Step Solution

Answer

The correct answer is d. 13.

Step 1: Identify vectors' components

The vectors \(\mathbf{U}\) and \(\mathbf{V}\) are given as \(\mathbf{U} = 7\mathbf{i} + 2\mathbf{j}\) and \(\mathbf{V} = 3\mathbf{i} - 4\mathbf{j}\). Here, the components are \(u_1 = 7\), \(u_2 = 2\), \(v_1 = 3\), and \(v_2 = -4\).

Step 2: Apply the dot product formula

The dot product of two vectors \(\mathbf{U}\) and \(\mathbf{V}\) is given by the formula \(\mathbf{U} \cdot \mathbf{V} = u_1v_1 + u_2v_2\). Use this formula with the components identified in Step 1.

Step 3: Calculate the dot product

Substitute the components into the formula: \[\mathbf{U} \cdot \mathbf{V} = 7 \times 3 + 2 \times (-4)\]Simplify the expression:\[21 - 8 = 13\]

Step 4: Choose the correct answer

The calculated dot product is 13. Comparing this result with the provided answers, option (d) is correct.

Key Concepts

Vectors

Vectors are fundamental tools in mathematics and physics that represent quantities with both magnitude and direction. They are essential in describing physical phenomena such as forces and velocities. Instead of describing a position in terms of coordinates alone, vectors can capture not just the point, but also the motion or force applied in a certain direction. For example, the vector \(\mathbf{U} = 7\mathbf{i} + 2\mathbf{j}\) indicates a direction along the x-axis, represented by \(\mathbf{i}\), and a smaller direction along the y-axis, represented by \(\mathbf{j}\).
The use of unit vectors (\(\mathbf{i}\) and \(\mathbf{j}\)) helps to distinctly separate each dimension's contribution to the vector as much as possible. Overall, vectors are not just arrows on paper; they are powerful representations that handle various multi-dimensional problems in a neat form.

Vector Components

Breaking vectors into components is like translating a complex problem into simpler parts. Every vector can essentially be seen as a sum of its components along each dimension. For a 2D vector, like \(\mathbf{V} = 3\mathbf{i} - 4\mathbf{j}\), it has components of 3 along the x-axis and -4 along the y-axis.
This breakdown allows easier application of various operations using simpler arithmetic rules.

• x-axis component (in direction of \(\mathbf{i}\))
• y-axis component (in direction of \(\mathbf{j}\))

These components, \(v_1 = 3\) and \(v_2 = -4\), make computational tasks with vectors like addition, subtraction and dot product straightforward and logical.

Trigonometry Problem Solving

Trigonometry plays a crucial role when dealing with vectors, particularly in solving problems involving angles and magnitudes. However, in the realm of vectors, the dot product offers a true intersection between trigonometry and algebra. The dot product of two vectors can be calculated easily by:

• Multiplying the magnitudes of the components in the same direction
• Adding the results of those products

For \(\mathbf{U} = 7\mathbf{i} + 2\mathbf{j}\) and \(\mathbf{V} = 3\mathbf{i} - 4\mathbf{j}\), the calculation follows the formula: \(\mathbf{U} \cdot \mathbf{V} = u_1v_1 + u_2v_2\).
By applying the formula, you get \(7 \times 3 + 2 \times (-4) = 13\). This product is more than just a number; it reflects how much of one vector goes in the direction of the other, showing the interconnectedness of vectors, trigonometry, and algebra.