Vector Unit

Find the unit vector perpendicular to the plane formed by the two vectors: U = 2 i + 10 j and V = i + 3 j – 5 k

Expand Hint
$$$\vec{a} \times \vec{b}=\begin{bmatrix}a_2b_3-a_3b_2\\ a_3b_1-a_1b_3\\ a_1b_2-a_2b_1\end{bmatrix}$$$
Hint 2
$$$\vec{a}=(a_{1}, a_{2}, a_{3})$$$
$$$\vec{b}=(b_{1}, b_{2}, b_{3})$$$
This is a two part problem, where the perpendicular vector must be determined first. Then, the unit vector will be solved next. Recall that a unit vector is a vector that has a magnitude of 1. Finding the cross product determines the perpendicular vector of U and V. Using the shown formula to get started:
$$$\vec{a} \times \vec{b}=\begin{bmatrix}a_2b_3-a_3b_2\\ a_3b_1-a_1b_3\\ a_1b_2-a_2b_1\end{bmatrix}$$$
where $$\vec{a}=(a_{1}, a_{2}, a_{3})$$ and $$\vec{b}=(b_{1}, b_{2}, b_{3})$$ .

Thus, the cross product is:
$$$\vec{U} \times \vec{V}=\begin{bmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\ U_x & U_y & U_z\\ V_x & V_y & V_z\end{bmatrix}=\begin{bmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\ 2 & 10 & 0\\ 1 & 3 & -5\end{bmatrix}$$$
$$$=\mathbf{i}[(10)(-5)-(0)(3)] - \mathbf{j}[(2)(-5)-(0)(1)] + \mathbf{k}[(2)(3)-(10)(1)]$$$
$$$=\mathbf{i}(-50-0) - \mathbf{j}(-10-0) + \mathbf{k}(6-10)=\{-50;\:10;\:-4 \}$$$
To find the unit vector, divide by the magnitude. To find the magnitude:
$$$|\vec{a}|=\sqrt{a_{x}^{2}+a_{y}^{2}+a_{z}^{2}}=\sqrt{(-50)^2+10^2+(-4)^2}=\sqrt{2,500+100+16}$$$
$$$=\sqrt{2,616}\approx 51.15$$$
Finally, the unit vector perpendicular to the plane formed by vectors U and V :
$$$\frac{-50i+10j-4k}{51.15}$$$
$$$\frac{-50i+10j-4k}{51.15}$$$
Time Analysis See how quickly you looked at the hint, solution, and answer. This is important for making sure you will finish the FE Exam in time.
  • Hint: Not clicked
  • Solution: Not clicked
  • Answer: Not clicked

Similar Problems from FE Sub Section: Vectors
049. Cross Products
050. Unit Vectors
306. Dot Product
419. Dot Prod
424. Vector Dot Product
535. X Product
543. Cross vs Dot Product
639. Vector Combinations

Similar Problems from FE Section: Determinants
049. Cross Products
050. Unit Vectors
123. Matrix Determinant
251. Matrix
252. Larger Determinant
256. 4x4 Matrix
306. Dot Product
419. Dot Prod
424. Vector Dot Product
447. 3x3 Matrix
503. Determinant Matrix
504. 3x3 Determinant
535. X Product
543. Cross vs Dot Product
633. 2x2 Matrix
639. Vector Combinations