Space vectors 3D

The vectors u = (1; 3; -4), v = (0; 1; 1) are given. Find the size of these vectors, calculate the angle of the vectors, the distance between the vectors.

Correct answer:

a =  5.099
b =  1.4142
A =  13.8979 °
h =  5.4772

Step-by-step explanation:

$$\begin{gathered} u=(1,3,-4) \\ v=(0,1,1) \\ a=\mathrm{\mid }u\mathrm{\mid }=\sqrt{u_x^2+u_y^2+u_z^2}=\sqrt{1^2+3^2+(-4{)}^2}=\sqrt{26}=5.099 \end{gathered}$$

$b=\mathrm{\mid }v\mathrm{\mid }=\sqrt{v_x^2+v_y^2+v_z^2}=\sqrt{0^2+1^2+1^2}=\sqrt{2}=1.4142$

$$\begin{gathered} c={\displaystyle \frac{u_x\cdot v_x+u_y+v_y+u_y\cdot v_y}{a\cdot b}}={\displaystyle \frac{1\cdot 0+3+1+3\cdot 1}{5.099\cdot 1.4142}}\doteq 0.9707 \\ A={\displaystyle \frac{180^{\circ }}{\pi }}\cdot \arccos (c)={\displaystyle \frac{180^{\circ }}{\pi }}\cdot \arccos (0.9707)=13.8979^{\circ }=13^{\circ }53^{\mathrm{\prime }}52\mathrm{"} \end{gathered}$$

$$\begin{gathered} w=u-v \\ w=(1,2,-5) \\ h=\mathrm{\mid }w\mathrm{\mid }=\sqrt{w_x^2+w_y^2+w_z^2}=\sqrt{1^2+2^2+(-5{)}^2}=\sqrt{30}=5.4772 \end{gathered}$$

Try another example

Related math problems and questions:

• Angle of the body diagonals
  Using vector dot product calculate the angle of the body diagonals of the cube.
• Angle between vectors
  Find the angle between the given vectors to the nearest tenth of a degree. u = (-22, 11) and v = (16, 20)
• Cuboids
  Two separate cuboids with different orientation in space. Determine the angle between them, knowing the direction cosine matrix for each separate cuboid. u1=(0.62955056, 0.094432584, 0.77119944) u2=(0.14484653, 0.9208101, 0.36211633)
• Triangle
  Plane coordinates of vertices: K[11, -10] L[10, 12] M[1, 3] give Triangle KLM. Calculate its area and its interior angles.
• Vector v4
  Find the vector v4 perpendicular to vectors v1 = (1, 1, 1, -1), v2 = (1, 1, -1, 1) and v3 = (0, 0, 1, 1)
• Parametric form
  Calculate the distance of point A [2,1] from the line p: X = -1 + 3 t Y = 5-4 t Line p has a parametric form of the line equation. ..
• Find the 5
  Find the equation of the circle with center at (1,20), which touches the line 8x+5y-19=0
• Find the 10
  Find the value of t if 2tx+5y-6=0 and 5x-4y+8=0 are perpendicular, parallel, what angle does each of the lines make with the x-axis, find the angle between the lines?
• Vectors abs sum diff
  The vectors a = (4,2), b = (- 2,1) are given. Calculate: a) |a+b|, b) |a|+|b|, c) |a-b|, d) |a|-|b|.
• Vector sum
  The magnitude of the vector u is 12 and the magnitude of the vector v is 8. Angle between vectors is 61°. What is the magnitude of the vector u + v?
• Scalar product
  Calculate the scalar product of two vectors: (2.5) (-1, -4)
• Scalar dot product
  Calculate u.v if |u| = 5, |v| = 2 and when angle between the vectors u, v is: a) 60° b) 45° c) 120°
• On a line
  On a line p : 3 x - 4 y - 3 = 0, determine the point C equidistant from points A[4, 4] and B[7, 1].
• Three points
  Three points K (-3; 2), L (-1; 4), M (3, -4) are given. Find out: (a) whether the triangle KLM is right b) calculate the length of the line to the k side c) write the coordinates of the vector LM d) write the directional form of the KM side e) write the d
• Distance of points
  A regular quadrilateral pyramid ABCDV is given, in which edge AB = a = 4 cm and height v = 8 cm. Let S be the center of the CV. Find the distance of points A and S.
• Rectangle 39
  Find the perimeter and area of the rectangular with vertices (-1, 4), (0,4), (0, -1), and (-4, 4)
• Perpendicular projection
  Determine the distance of a point B[1, -3] from the perpendicular projection of a point A[3, -2] on a straight line 2 x + y + 1 = 0.