Problem 29 - Precalculus Mathematics for Calculus

Question

Determine whether or not the given vectors are perpendicular. $$(4,-2,-4),\langle 1,-2,2\rangle$$

Step-by-Step Solution

Answer

The vectors are perpendicular because their dot product is zero.

Step 1: Understand the Perpendicular Condition

Two vectors are perpendicular if their dot product equals zero. This is because the dot product is related to the cosine of the angle between the vectors, and when vectors are perpendicular, the angle is 90 degrees, making the cosine zero.

Step 2: Write the Dot Product Formula

To find the dot product of two vectors, use the formula:\[ a \cdot b = a_xb_x + a_yb_y + a_zb_z \]where the vectors are \( a = (a_x, a_y, a_z) \) and \( b = (b_x, b_y, b_z) \).

Step 3: Substitute the Given Vectors

Substitute the given vectors into the dot product formula:\[(4,-2,-4) \cdot (1,-2,2) = 4 \times 1 + (-2) \times (-2) + (-4) \times 2 \] which simplifies to \[ 4 + 4 - 8 \].

Step 4: Calculate the Dot Product

Now, calculate the expression:\[ 4 + 4 - 8 = 0 \]. Since the result is zero, it means the vectors are perpendicular.

Key Concepts

Dot Product

The dot product is a fundamental operation involving two vectors. It provides a single scalar value, summarizing how much one vector extends in the direction of another. To compute the dot product between two vectors, you simply multiply corresponding components and then sum the results. This operation can be expressed with the formula:
\[ a \cdot b = a_xb_x + a_yb_y + a_zb_z \]
where \(a\) and \(b\) are vectors with components \((a_x, a_y, a_z)\) and \((b_x, b_y, b_z)\) respectively.

Understanding the properties of the dot product is crucial. Here are some key points:

• If the dot product is positive, the vectors are in a similar direction.
• If the dot product equals zero, the vectors are perpendicular.
• If the dot product is negative, the vectors are in opposite directions.

Grasping these properties makes it easier to determine the relationship between vectors.

Vectors

Vectors are essential in various fields such as physics, engineering, and mathematics. They are used to represent quantities that have both magnitude and direction. A vector can be visualized as an arrow pointing from one point to another in space.

Mathematically, vectors are often denoted using

• parentheses like \((x, y, z)\), or
• angle brackets like \(~\langle x, y, z \rangle ~\)

These notations indicate the components of the vector in three-dimensional space.

There are some fundamental operations associated with vectors:

• Addition: Combining vectors by adding their corresponding components.
• Scalar Multiplication: Multiplying a vector by a scalar to stretch or shrink it.
• Dot Product: As previously discussed, producing a scalar that represents the relationship between vectors.

These operations enable us to perform complex calculations and analyze problems involving multiple directions.

Perpendicular Vectors

For two vectors to be perpendicular, they must form a 90-degree angle. This concept is not only visual but also has a mathematical foundation.

In vector algebra, the condition for two vectors \(a\) and \(b\) to be perpendicular is that their dot product is zero. This condition is expressed as:
\[ a \cdot b = 0 \]
When the dot product is zero, it implies that the angle between the vectors is 90 degrees, or \( \frac{ \pi }{2} \) radians, where the cosine of this angle is zero.

Understanding perpendicularity has practical implications:

• In physics, it explains orthogonal forces that don't affect each other's magnitude.
• In geometry, it's used to determine right angles in multi-dimensional space.

By understanding when vectors are perpendicular, you can analyze and solve problems in both theoretical and practical applications efficiently.