Dot Product Is Zero: Find Possible Vectors U & V

In linear algebra, the dot product is a fundamental operation that provides valuable insights into the relationship between vectors. Specifically, if the dot product of two vectors is zero, it implies that the vectors are orthogonal (perpendicular) to each other. This concept is crucial in various applications, including physics, engineering, and computer graphics. Let's delve into the specifics of how to determine which vectors satisfy this condition. Guys, understanding this relationship is super important for grasping more complex concepts later on!

Understanding the Dot Product

Before we dive into the options, let's quickly recap what the dot product actually is. Given two vectors, ${ u = \langle u_1, u_2 \rangle }$ and ${ v = \langle v_1, v_2 \rangle }$, their dot product, denoted as ${ u \cdot v }$, is calculated as:

${ u \cdot v = u_1v_1 + u_2v_2 }$

The result of the dot product is a scalar (a single number), not another vector. The dot product tells us something about the angle between the two vectors. If the dot product is zero, the cosine of the angle between the vectors is zero, which means the angle is 90 degrees (or ${ \pi/2 }$ radians). Basically, they're perpendicular!

When the dot product of two vectors equals zero, it indicates that the vectors are orthogonal. Orthogonality is a key concept in various fields, including linear algebra, calculus, and physics. For example, in physics, if two forces acting on an object are orthogonal, the work done by one force does not contribute to the work done by the other force. In linear algebra, orthogonal vectors form a basis for vector spaces and are used in techniques such as Gram-Schmidt orthogonalization to simplify calculations and solve systems of equations. Understanding orthogonality also helps in solving problems involving projections, where one vector is projected onto another. By using orthogonal vectors, you can break down complex vector components into simpler, independent parts. This simplifies the analysis and solution of many problems.

Analyzing the Given Options

Now, let's examine each option provided to determine which pair of vectors results in a dot product of zero.

A. ${ u = \langle 3, 4 \rangle }$ and ${ v = \langle 4, -3 \rangle }$ B. ${ u = \langle 3, 4 \rangle }$ and ${ v = \langle -4, -3 \rangle }$ C. ${ u = \langle -3, 4 \rangle }$ and ${ v = \langle 4, -3 \rangle }$ D. ${ u = \langle -3, -4 \rangle }$ and ${ v = \langle 4, -3 \rangle }$

Option A: u = <3, 4> and v = <4, -3>

Let's calculate the dot product for option A:

${ u \cdot v = (3)(4) + (4)(-3) = 12 - 12 = 0 }$

Since the dot product is 0, these vectors are orthogonal. This means option A is a possible solution.

To verify that option A, ${ u = \langle 3, 4 \rangle }$ and ${ v = \langle 4, -3 \rangle }$, results in a dot product of zero, we can perform the dot product calculation:

${ u \cdot v = (3 \times 4) + (4 \times -3) = 12 - 12 = 0 }$

Since the dot product is indeed zero, this confirms that vectors ${ u }$ and ${ v }$ are orthogonal. Orthogonal vectors are perpendicular to each other, meaning they form a 90-degree angle. This property is widely used in physics, engineering, and computer graphics for various calculations and transformations. For example, in computer graphics, orthogonal vectors can be used to define coordinate systems, making it easier to manipulate and render 3D objects.

Option B: u = <3, 4> and v = <-4, -3>

Now, let's check option B:

${ u \cdot v = (3)(-4) + (4)(-3) = -12 - 12 = -24 }$

The dot product is -24, which is not 0. Thus, these vectors are not orthogonal.

To further elaborate, the dot product of option B, where ${ u = \langle 3, 4 \rangle }$ and ${ v = \langle -4, -3 \rangle }$, is calculated as follows:

${ u \cdot v = (3 \times -4) + (4 \times -3) = -12 - 12 = -24 }$

Since the result is -24 and not zero, vectors ${ u }$ and ${ v }$ are not orthogonal. This indicates that these vectors do not meet the condition of being perpendicular to each other. A non-zero dot product implies that the vectors have some degree of alignment or opposition. In this case, the negative dot product suggests that the vectors are somewhat opposed in direction.

Option C: u = <-3, 4> and v = <4, -3>

Next, let's evaluate option C:

${ u \cdot v = (-3)(4) + (4)(-3) = -12 - 12 = -24 }$

Again, the dot product is -24, not 0. These vectors are not orthogonal.

For option C, where ${ u = \langle -3, 4 \rangle }$ and ${ v = \langle 4, -3 \rangle }$, the dot product is:

${ u \cdot v = (-3 \times 4) + (4 \times -3) = -12 - 12 = -24 }$

As the dot product is -24, which is not zero, vectors ${ u }$ and ${ v }$ are not orthogonal. This means they are not perpendicular to each other. Understanding this helps in discerning the relationships between vectors in various contexts, such as determining force components or analyzing geometric configurations. In applications where orthogonality is required, such as basis transformations, these vectors would not be suitable.

Option D: u = <-3, -4> and v = <4, -3>

Finally, let's consider option D:

${ u \cdot v = (-3)(4) + (-4)(-3) = -12 + 12 = 0 }$

Since the dot product is 0, these vectors are orthogonal. This indicates option D is another possible solution.

To verify that option D, ${ u = \langle -3, -4 \rangle }$ and ${ v = \langle 4, -3 \rangle }$, results in a dot product of zero, we can calculate:

${ u \cdot v = (-3 \times 4) + (-4 \times -3) = -12 + 12 = 0 }$

Since the dot product equals zero, this confirms that vectors ${ u }$ and ${ v }$ are orthogonal. This orthogonality implies that these vectors are perpendicular to each other. In practical applications, orthogonal vectors are invaluable for simplifying complex calculations and providing clear insights into vector relationships.

Conclusion

Based on our calculations, options A and D both satisfy the condition that ${ u \cdot v = 0 }$. Therefore, the vectors in options A and D could represent ${ u }$ and ${ v }$. Remember guys, orthogonality is key!

Final Answer: The final answer is ${ A }$ and ${ D }$