Find a unit vector orthogonal to $3 \mathbf { i } + 2 \mathbf { j }$ and $\mathbf { j } + 5 \mathbf { k }$.
A) $10 \mathbf { i } - 15 \mathbf { j } + 3 \mathbf { k }$
B) $\frac { 10 } { \sqrt { 334 } } \mathbf { i } - \frac { 15 } { \sqrt { 334 } } \mathbf { j } + \frac { 3 } { \sqrt { 334 } } \mathbf { k }$
C) $10 \mathbf { i } + 15 \mathbf { j } + 3 \mathbf { k }$
D) $- \frac { 10 } { \sqrt { 334 } } \mathbf { i } + \frac { 15 } { \sqrt { 334 } } \mathbf { j } - \frac { 3 } { \sqrt { 334 } } \mathbf { k }$
E) $\frac { 10 } { \sqrt { 334 } } \mathbf { i } + \frac { 15 } { \sqrt { 334 } } \mathbf { j } + \frac { 3 } { \sqrt { 334 } } \mathbf { k }$

Question

Quizplus · Accepted Answer

To find a vector orthogonal to both $3 \mathbf { i } + 2 \mathbf { j }$ and $\mathbf { j } + 5 \mathbf { k }$, we can use the cross product of these two vectors. The cross product is calculated as:$$\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \3 & 2 & 0 \0 & 1 & 5 \\end{vmatrix}= (10 \mathbf { i } - 15 \mathbf { j } + 3 \mathbf { k })$$To make it a unit vector, we divide by its magnitude:$$\sqrt{10^2 + (-15)^2 + 3^2} = \sqrt{100 + 225 + 9} = \sqrt{334}$$Thus, the unit vector is:$$\frac { 10 } { \sqrt { 334 } } \mathbf { i } - \frac { 15 } { \sqrt { 334 } } \mathbf { j } + \frac { 3 } { \sqrt { 334 } } \mathbf { k }$$

Find a Unit Vector Orthogonal To 3i+2j3 \mathbf { i } + 2 \mathbf { j }3i+2j

Find a Unit Vector Orthogonal To $3 \mathbf { i } + 2 \mathbf { j }$