Question

The dot product of two vectors

x=[x1] and y = [y1]
[x2] [y2]
[ ⋮ ] [ ⋮ ]
[xn] [yn]
in R" is defined by x.y=x1y1+x2y2+...+xnyn. The vectors 7 and y are called perpendicular if y=0. Any vector in Rº perpendicular to
[-5]
[-4]
[-7]
can be written in the form
[ ] a + [ ] t
[ ] [ ]
[ ] [ ]

Answer 1

Vectors x and y are perpendicular if their dot product is zero. A vector perpendicular to
`\( \begin{bmatrix} -5 \\ -4 \\ -7 \end{bmatrix} \)` can be expressed as
`\( \begin{bmatrix} -4 \\ 5 \\ 0 \end{bmatrix}t + \begin{bmatrix} -7 \\ 0 \\ 1 \end{bmatrix}s \).`

The dot product of two vectors
`\( x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} \)` and
`\( y = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix} \)` in
`\( \mathbb{R}^n \)` is defined as
`\( x \cdot y = x_1y_1 + x_2y_2 + \ldots + xnyn \).`

Two vectors x and y are called perpendicular if their dot product is zero, i.e.,
`\( x \cdot y = 0 \)`. In this case,
`\( x \cdot y = x_1y_1 + x_2y_2 + \ldots + xnyn = 0 \).`

Now, consider the vector
`\( y = \begin{bmatrix} -5 \\ -4 \\ -7 \end{bmatrix} \)` . A vector a is perpendicular to y if
`\( a \cdot y = 0 \)`. Let
`\( a = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \)`. Then, the condition for perpendicularity is:

`\[ a \cdot y = a_1(-5) + a_2(-4) + a_3(-7) = 0 \]`

This simplifies to
`\( -5a_1 - 4a_2 - 7a_3 = 0 \).`

The solution to this equation gives the set of all vectors a perpendicular to y in
`\( \mathbb{R}^3 \).` It is given by:
`\[ a = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} = \begin{bmatrix} -4 \\ 5 \\ 0 \end{bmatrix}t + \begin{bmatrix} -7 \\ 0 \\ 1 \end{bmatrix}s \]`where t and s are scalar parameters.

The question probable may be:

The dot product of two vectors
`\( x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} \)` and
`\( y = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix} \)` in R" is defined by
`\( x \cdot y = x_1y_1 + x_2y_2 + \ldots + xnyn \).` The vectors 7 and y are called perpendicular if y=0. Any vector in Rº perpendicular to
`\( \begin{bmatrix} -5 \\ -4 \\ -7 \end{bmatrix} \)` can be written in what the form?