Question

Given u = -5i + 8j and v= 56i + 35j, are u and v parallel or orthogonal? Explain.

Answer 1

Perpendicular

Given the following vectors:

`\vec u =-5 \hat i +8 \hat j\\\vec v =56 \hat i +35 \hat j\\`

We are asked to determine whether vectors u and v are parallel or perpendicular.

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How do we do this?

To determine whether two vectors are parallel, the result of their cross product is zero.

To determine whether two vectors are perpendicular, the result of their dot product is zero.

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`\bold{Given \ two \ vectors...}\\\vec a=a_x \hat i + a_y \hat j +a_z \hat k\\\\\vec b=b_x \hat i + b_y \hat j +b_z \hat k\\`

`\bold{Cross \ Product} \Rightarrow \vec a * \vec b= \begin{vmatrix}\hat i & \hat j & \hat k\\a_x & a_y & a_z\\b_x & b_y & b_z\end{vmatrix} \rightarrow \begin{vmatrix}a_y & a_z \\b_y & b_z \end{vmatrix} \hat i- \begin{vmatrix}a_x & a_z \\b_x & b_z \end{vmatrix} \hat j+\begin{vmatrix}a_x & a_y \\b_x & b_y \end{vmatrix} \hat k`

`\bold{Dot \ Product} \Rightarrow \vec a \cdot \vec b = a_xb_x+a_yb_y+a_zb_z`

Note: The result of the cross product is a vector while the result of a dot product is a scalar.

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Cross product:

`\vec u * \vec v = \begin{vmatrix}\hat i & \hat j & \hat k\\-5 & 8 & 0\\56 & 35 & 0\end{vmatrix}`

`\Longrightarrow \begin{vmatrix} 8 & 0 \\ 35 & 0 \end{vmatrix} \hat i- \begin{vmatrix}-5 & 0 \\56 & 0 \end{vmatrix} \hat j+\begin{vmatrix}-5 & 8 \\56 & 35 \end{vmatrix} \hat k`

`\Longrightarrow [(8)(0)-(0)(35)]\hat i -[(-5)(0)-(0)(56)]\hat j +[(-5)(35)-(8)(56)] \hat k`

`\Longrightarrow (0)\hat i (0)\hat j +[-175-448] \hat k`

`\Longrightarrow (0)\hat i (0)\hat j +(-623) \hat k`

`Thus, \ \boxed{\vec u * \vec v = 0\hat i + 0\hat j -623 \hat k}`

Which does not equal zero. So, these vectors aren't parallel.

Now, dot product:

`\vec u \cdot \vec v = (-5)(56)+(8)(35)+(0)(0)`

`\Longrightarrow \vec u \cdot \vec v = -280+280+0`

`Thus, \ \boxed{ \vec u \cdot \vec v = 0}`

The dot product of vectors u and v equals zero. These vectors are perpendicular!