Find the following: 1) The area of the triangle whose vertices are the points P(1,0,2), Q(3, -1,3), and R(4,1,2). 2) Parametric equations of the line passing through the point P(2,1, -3) and parallel to the vector v = 2i + j - k. 3) Parametric equations of the line passing through the points P(2, -2,3) and Q(2,1, -2). 4) An equation of the of the plane passing through the point P(1,0, -2) normal to the vector n = ?2, -3,1?. 5) An equation of the of the plane passing through the points P(0,2,1), Q(2,1,2), and R(3,3,1). 6) The distance from the point S(2, -3,1) to the line L: x = 1 - t, y = 2 + 3t, z = 5 + 2t. 7) The distance from the point S(1,0, -2) to the plane 2x - 3y + z = 12. 8) The point of intersection of the line L: x = 1 + t, y = 2 - 3t, z = 5 + 2t and the plane 2x + y - 3z = 3. 9) The line of intersection of the two planes 2x + y - z = 1 and x - y + 3z = 2. 10) The angle between the planes 2x + 3y - z = 9 and x + 5y + 3z = 2.
Added by Samantha R.
Close
Step 1
Step 1: Find the vectors representing the sides of the triangle: The vector representing side PQ is Q - P = (3-1, -1-0, 3-2) = (2, -1, 1) The vector representing side PR is R - P = (4-1, 1-0, 2-2) = (3, 1, 0) Show more…
Show all steps
Your feedback will help us improve your experience
Sri K and 50 other Calculus 1 / AB educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Find parametric and symmetric equations for the line containing the point P(-4,2,1) that is parallel to the line with parametric equation: x = -1 + 4t, y = 3 - 2t, z = 1 + 3t Find the vector equation, parametric and symmetric equations for the line through the point P(5,1,3) that is parallel to 2i-3j+4k Find an equation of the plane containing the point (1,2,3) that is perpendicular to the line with parametric equations x = -1 - 4t, y = 2 - 3t, z = 2 + t Find an equation of the plane containing the points P(-1,2,3), Q(2,1,3) and R(-3,3,2). Sketch the plane in the first octant. Determine whether the lines are parallel, skew or intersecting. If they intersect, find the point of intersection L1: x = 1 + 3t, y = 4 - 2t, z = -2 + 2t L2: x = 6 + s, y = -3 + 3s, z = 2s Name and sketch the surfaces in three dimensions.
Sri K.
Column 1 (a) The centroid of the triangle with vertices at the points where the plane $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}$ $=1$ meets thecoordinate axes is $\left(1, r, r^{2}\right)$. The plane passes through the point $(4,-8,15)$ if $r$ is cqual to (b) The line $\frac{x-1}{r}=\frac{y+2}{r^{2}}$ $=\frac{z}{-12}$ is parallel to the plane $x+y+z=3$ if $r$ is equal to (c) The plane is perpendicular to the line $\frac{x}{1}=\frac{y}{r}=\frac{z}{r^{2}}$ passes through the origin and the point $(-4,3,1)$ if $r$ is equal to (d) The lines whose vectorial cquation are $\mathbf{r}=2 \mathbf{i}-3 \mathbf{j}$ $+7 x+\lambda(5 \mathbf{i}+p \mathbf{j}+2 p \mathbf{k})$ and $\mathbf{r}=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}+\mu$ $(-i-p \mathbf{j}+3 \mathbf{k})$ are perpendicular for all values of $\lambda$ and $\mu$ if $p$ is equal to Column 2 (p) 1 (q) 5 (r) $-4$ (s) 3
Let $\mathcal{L}$ denote the intersection of the planes $x-y-z=1$ and $2 x+3 y+z=2 .$ Find parametric equations for the line $\mathcal{L} .$ Hint: To find a point on $\mathcal{L},$ substitute an arbitrary value for $z(\mathrm{say}, z=2)$ and then solve the resulting pair of equations for $x$ and $y .$
VECTOR GEOMETRY
Planes in 3 -Space
Recommended Textbooks
Calculus: Early Transcendentals
Thomas Calculus
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD