Question

Given that \(p=2 i+2 j\) and \(q=7 k\) find \(p \cdot q\) and interpret this result geometrically.

Short answer

Based on the dot product result, the geometric interpretation of the relationship between vectors p and q is that they are orthogonal (perpendicular) to each other, forming a 90-degree angle in the geometric space.

Step-by-step solution

Write down the given vectors

We are given two vectors p and q: \(p = 2 i + 2 j\) and \(q = 7 k\).

Compute the dot product

To compute the dot product of vectors p and q, we multiply their corresponding components and add them up: \(p \cdot q = (2 i + 2 j) \cdot (7 k) = 2 \cdot 0 + 2 \cdot 0 + 0 \cdot 7 = 0\)

Interpret the result geometrically

Since the dot product \(p \cdot q = 0\), this implies that the angle between vectors p and q is 90 degrees (since the dot product \(a \cdot b = |a||b|\cos(\theta)\) , and if \(\theta=90^\circ\), then \(\cos(90^\circ)=0\)). Therefore, the given vectors p and q are orthogonal (perpendicular) to each other in the geometric space.

Key concepts

Geometric Interpretation of the Dot Product

When dealing with vectors, it can be incredibly helpful to understand them not just in terms of their algebraic properties, but geometrically as well. The dot product, also known as the scalar product, is one of those interactions between vectors that provide clear geometric insight.

The dot product of two vectors results in a scalar, which can tell us about the angle between the two vectors. The formula for the dot product is \(a \cdot b = |a||b|\cos(\theta)\), where \(|a|\) and \(|b|\) are the magnitudes (or lengths) of vectors \(a\) and \(b\), and \(\theta\) is the angle between them.

• If the dot product is positive, the angle is acute.
• If it is zero, the angle is 90 degrees, meaning the vectors are perpendicular, or orthogonal.
• If it is negative, the vectors form an obtuse angle.

Thus, the dot product informs us not just that two vectors interact, but how they align or oppose in the space.

Orthogonal Vectors

Orthogonal vectors are fascinating in their properties. Two vectors that are orthogonal form a right angle, meaning they are perpendicular to each other. For two vectors to be orthogonal, their dot product must equal zero. This relationship is vital in many mathematical spaces and applications.

Why is this important? In vector geometry and linear algebra, detecting orthogonal vectors helps simplify problems. For example:

• They can be used to define orthogonal coordinate systems, like the familiar Cartesian coordinate system, where axes are perpendicular to one another.
• In physics, orthogonal vectors represent independent forces or movements, which don't interfere with each other.
• This concept is also pivotal in computer graphics, where orthogonality ensures proper representation of dimensions.

Knowing that \(p \cdot q = 0\) directly indicates orthogonality helps us quickly assess that vectors \(p\) and \(q\) are perpendicular, as seen in the original problem.

Understanding Vector Components

Vectors in 3D space are expressed in terms of components aligning with the coordinate axes, usually denoted by \(i, j,\) and \(k\). These basis vectors show movement along the x, y, and z axes respectively. Any vector can be broken down into its components in these directions.

For instance, in the given vectors \(p = 2i + 2j\) and \(q = 7k\), each is expressed in terms of its components along the axes:

• Vector \(p\) has components only along the x (2) and y axes (2), with no z component (0).
• Vector \(q\) moves solely along the z axis (7), with no x or y components (0).

These components are treated as separate influences along the axes and can greatly simplify calculations.

When finding the dot product, each corresponding component pair is multiplied and summed. If vectors share no common dimensions in space (like \(p\) and \(q\)), their dot product will naturally result in zero, confirming their orthogonality. Understanding these components keeps vector manipulations intuitive and ensures accurate geometric interpretations.