91Ó°ÊÓ

The dot product, also known as the scalar product, is a crucial concept in vector mathematics, especially when determining if two vectors are perpendicular. Two vectors are perpendicular if their dot product is zero. For vectors \( \mathbf{u} = i + mj + k \) and \( \mathbf{v} = 2i - j + nk \), their dot product is calculated as follows:

• Multiply the corresponding components of the vectors: \( 1 \cdot 2 \) for the \(i\) components, \( m \cdot (-1) \) for the \(j\) components, and \( 1 \cdot n \) for the \(k\) components.
• Add these products together: \( 2 - m + n \).

The equation \( 2 - m + n = 0 \) must hold true for the vectors to be perpendicular, making this equation our first step in solving such a problem. This understanding helps in visualizing how changes to the parameters \( m \) and \( n \) will affect the relationship between \( \mathbf{u} \) and \( \mathbf{v} \).

The concept of magnitude equality between two vectors means that both vectors have the same length. To find the magnitude of a vector, calculate the square root of the sum of the squares of its components. In our example, the magnitudes of vectors \( \mathbf{u} = i + mj + k \) and \( \mathbf{v} = 2i - j + nk \) are:

• \( |\mathbf{u}| = \sqrt{1^2 + m^2 + 1^2} = \sqrt{2 + m^2} \)
• \( |\mathbf{v}| = \sqrt{2^2 + (-1)^2 + n^2} = \sqrt{5 + n^2} \)

Magnitude equality for these vectors means \( \sqrt{2 + m^2} = \sqrt{5 + n^2} \). By squaring both sides (to remove the square roots), we arrive at the equation \( 2 + m^2 = 5 + n^2 \). This gives us a second essential condition to use in finding the unknown variables \( m \) and \( n \). Understanding how magnitudes relate provides insight into the physical properties and comparisons of vectors.

When tasked with solving equations derived from vector properties, breaking down the process into manageable steps is essential. Here, we have two equations:

• \( 2 - m + n = 0 \), from the dot product condition
• \( 2 + m^2 = 5 + n^2 \), from the magnitude equality condition

Start by solving one equation for one variable. Let's solve the dot product equation for \( n \):

• From \( 2 - m + n = 0 \), we find \( n = m - 2 \).

Substitute \( n = m - 2 \) into the second equation:

• \( 2 + m^2 = 5 + (m-2)^2 \)

Expand the expression \( (m-2)^2 \) to get \( m^2 - 4m + 4 \), which simplifies the equation to:

• \( 2 + m^2 = 5 + m^2 - 4m + 4 \)

After cancelling \( m^2 \) from both sides, simplify to find \( 4m = 7 \), leading to \( m = \frac{7}{4} \). Substitute back this value into the equation for \( n \) to get \( n = \frac{-1}{4} \). This process illustrates how a systematic approach to solving equations can reveal the undefined variables with precision.