Question

Find a vector orthogonal to both <-3,-2,0> and <0,-2,-5> to the form <1,,>

Answer 1

To find a vector orthogonal to both <-3,-2,0> and <0,-2,-5>, we can take their cross product. The cross product of the given vectors is <1,-10,15>.

Step-by-step explanation:

To find a vector orthogonal to both \(\langle -3,-2,0 \rangle\) and \(\langle 0,-2,-5 \rangle\) in the form \(\langle 1,x,y \rangle\), we can take their cross product.

The cross product of two vectors, \(\vec{u}\) and \(\vec{v}\), can be found using the formula:

\(\vec{u} \times \vec{v} = \langle u_yv_z - u_zv_y, u_zv_x - u_xv_z, u_xv_y - u_yv_x \rangle\)

Plugging in the given vectors:

\(\langle -3,-2,0 \rangle \times \langle 0,-2,-5 \rangle = \langle (-2)(-5) - 0(-2), 0(0) - (-3)(-5), (-3)(-2) - (-2)(0) \rangle\)

Simplifying the calculations gives us:

\(\langle -3,-2,0 \rangle \times \langle 0,-2,-5 \rangle = \langle -10,15,-6 \rangle\)

Therefore, the vector \(\langle 1,-10,15 \rangle\) is orthogonal to both \(\langle -3,-2,0 \rangle\) and \(\langle 0,-2,-5 \rangle\).

Answer 2

The vector < 1, -4, 2 > is orthogonal to both < -3, -2, 0 > and < 0, -2, -5 > because the dot product of < 1, -4, 2 > with each of the given vectors is zero.

Step-by-step explanation:

To find a vector orthogonal to two given vectors, we can use the cross product. Let the vectors be A = < a₁, a₂, a₃ > and B = < b₁, b₂, b₃ >. The cross product, C, is given by:

`\[ C = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a₁ & a₂ & a₃ \\ b₁ & b₂ & b₃ \end{vmatrix} \]`

For the given vectors A = < -3, -2, 0 > and B = < 0, -2, -5 >, the cross product C would be:

`\[ C = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ -3 & -2 & 0 \\ 0 & -2 & -5 \end{vmatrix} \]`

Expanding this determinant, we get:

`\[ C = \hat{i}(10) - \hat{j}(15) + \hat{k}(6) \]`

Therefore, C = < 10, -15, 6 >. This vector is orthogonal to both A and B. However, to match the specified form < 1, , >, we can multiply C by a scalar. In this case, multiplying by -1/15 gives us the final answer:

`\[ \text{Final Answer: } (-1)/(15) * < 10, -15, 6 > = < 1, -4, 2 > \]`

Thus, < 1, -4, 2 > is a vector orthogonal to both < -3, -2, 0 > and < 0, -2, -5 > in the specified form.