7.5: Equations of Lines and Planes in Space
- Page ID
- 143793
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- Consider the line \(\mathbf{r}(t) = (-5, -1) + t(2, 3)\). Find a point on the line and a vector parallel to the line.
- Consider the line described by the parametric equations \(x = 1 + 5t, y = 5 + 2t, z = 5\). Find a point on the line and a vector parallel to the line.
- Consider the line \(\mathbf{r} = t\mathbf{i} + (4 - 2t)\mathbf{j} + 3\mathbf{k}\). Find a point on the line and a vector parallel to the line.
- Find the equation of the line passing through the points \(P(4, -1)\) and \(Q(5, -3)\).
- Find the equation of the line passing through the points \(A(1, -4, -2)\) and \(B(-1, 0, 4)\).
- Find the equation of the line passing through the points \(S(2, 3, -2)\) and \(T(-3, 1, 2)\).
- Find the equation of the line passing through point \(P(1, 1)\) and orthogonal to vector \(\mathbf{v} = (5, -1)\).
- Parameterize the equation of a line \(\mathbf{r}(t)\) so that \(\mathbf{r}(0) = (-4, -2, -4)\) and \(\mathbf{r}(1) = (0, -5, 1)\).
- Find the distance from line \(\mathbf{r} = (4\mathbf{i} + 2\mathbf{j} + 5\mathbf{k}) + t(3\mathbf{i} + 2\mathbf{k})\) to point \(Q(2, -4, 0)\).
- Find the distance from line \(\mathbf{r} = (-1 - 2t, -3 - 5t)\) to point \(P(-4, 0)\).
- Determine if lines \(L_1: x = 1 + 3s, y = 5 - 2s\) and \(L_2: x = 2 - 9t, y = 6t\) are equal, parallel, skew, or intersecting.
- Determine if lines \(\mathbf{r}_1 = (-2, -2, -4) + s(1, 4, 3)\) and \(\mathbf{r}_2 = (-5, 2, -4) + t(-4, -2, 4)\) are equal, parallel, skew, or intersecting.
- Determine if the lines \(\mathbf{r}_1(s) = \mathbf{i} + (1 - 4s)\mathbf{j} + (4 + 5s)\mathbf{k}\) and \(\mathbf{r}_2(t) = (3 + 2t)\mathbf{i} + (4 - t)\mathbf{j} + (4 + 5t)\mathbf{k}\) are equal, parallel, skew, or intersecting.
- Determine if the lines \(L_1: x = -1 + 6s, y = 1 + 2s, z = 2 + 4s\) and \(L_2: x = 2 - 3t, y = 2 - t, z = 2 - 4t\) are equal, parallel, skew, or intersecting.
- Consider the plane \((2, 4, 0) \cdot \Bigl( (x, y, z) - (-1, 4, 5) \Bigr) = 0\). Find a point in the plane and a vector normal to the plane. Sketch the plane.
- Consider the plane \(5(x - 4) - 5(y - 5) + 3(z - 4) = 0\). Find a point in the plane and a vector normal to the plane. Sketch the plane.
- Consider the plane \( 2x - z = 4 \). Find a point in the plane and a vector normal to the plane. Sketch the plane.
- Find the equation of the plane passing through the points \(P(3, -2, -5)\), \(Q(-1, -2, 0)\), and \(R(-2, -1, 2)\).
- Find the equation of the plane passing through the points \(A(5, 3, 2)\), \(B(4, 3, -3)\), and \(C(-5, 2, 0)\).
- Find the equation of the plane passing through points \(P(-1, 0, 0)\) and \(Q(-1, 1, -1)\) and parallel to vector \(\mathbf{v} = \mathbf{i} + \mathbf{k}\).
- Find the distance from the plane \(-4(x - 5) - 2(z - 4) = 0\) to point \(A(3, 5, 2)\).
- Find the distance from the plane \(4x + 4y - 5z = -2\) to point \(B(2, 4, -3)\).
- Determine if the planes \(4x + 3y - 2z = 3\) and \(5x + y - z = 4\) intersect or are parallel. If the planes intersect, find the line of intersection.
- Determine if the planes \(3x - 2y -z = 0\) and \(-6x +4(y - 1) + 2(z + 4) = 0\) intersect or are parallel. If the planes intersect, find the line of intersection.
- Determine if the planes \(-4x - y - 3z = 2\) and \(4x + y - 3z = -2\) intersect or are parallel. If the planes intersect, find the line of intersection.
- Determine if the line \(\mathbf{r} = (-5, -4, -3) + t(2, -3, 2)\) and the plane \(2(x + 4) + 2(y + 4) + (z - 2) = 0\) intersect or are parallel. If they intersect, find the point of intersection.
- Determine if the line \(\mathbf{r} = -s\mathbf{i} - (3 + s)\mathbf{j} + (5 + s)\mathbf{k}\) and the plane \(-3(x + 4) - 5(y + 2) - (z - 1) = 0\) intersect or are parallel. If they intersect, find the point of intersection.