{"id":916,"date":"2025-10-14T15:55:51","date_gmt":"2025-10-14T15:55:51","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&p=916"},"modified":"2025-10-22T14:33:10","modified_gmt":"2025-10-22T14:33:10","slug":"lines-and-planes-in-space-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/lines-and-planes-in-space-learn-it-3\/","title":{"raw":"Lines and Planes in Space: Learn It 3","rendered":"Lines and Planes in Space: Learn It 3"},"content":{"raw":"

Relationships between Lines<\/h2>\r\n
\r\n

In two dimensions, two lines have only three possible relationships: they are equal (the same line), parallel but distinct, or they intersect at exactly one point. Three dimensions introduces a fourth possibility that doesn't exist in 2D geometry.<\/p>\r\n

When working with two lines in three-dimensional space, exactly one of these four relationships must be true:<\/p>\r\n\r\n

    \r\n \t
  1. Equal lines<\/strong>: The lines are identical (same direction, same points)<\/li>\r\n \t
  2. Parallel lines<\/strong>: The lines have the same direction but never meet<\/li>\r\n \t
  3. Intersecting lines<\/strong>: The lines cross at exactly one point<\/li>\r\n \t
  4. Skew lines<\/strong>: The lines are not parallel and do not intersect<\/li>\r\n<\/ol>\r\n
    \r\n

    skew lines<\/h3>\r\nTwo lines are skew if they are not parallel but do not intersect. This relationship is unique to three-dimensional space and cannot occur in a plane.\r\n\r\n<\/section>\r\n\r\n[caption id=\"attachment_5166\" align=\"aligncenter\" width=\"418\"]\"This Figure 5. In three dimensions, it is possible that two lines do not cross, even when they have different directions.[\/caption]\r\n\r\n<\/div>\r\n
    \r\n

    To classify the relationship between two lines, we need to answer two key questions:<\/p>\r\n\r\n

      \r\n \t
    1. Are the direction vectors parallel?<\/strong> (Do the lines have the same or opposite directions?)<\/li>\r\n \t
    2. Do the lines share any points?<\/strong> (Is there a common point on both lines?)<\/li>\r\n<\/ol>\r\nThe answers to these questions determine the relationship.\r\n\r\n<\/div>\r\n
      <\/div>\r\n
      \r\n\r\n[caption id=\"attachment_5167\" align=\"aligncenter\" width=\"610\"]\"This Figure 6. Determine the relationship between two lines based on whether their direction vectors are parallel and whether they share a point.[\/caption]\r\n\r\n
      \r\n

      line classification<\/h3>\r\n
        \r\n \t
      • Direction vectors parallel AND lines share a point<\/strong> \u2192 Lines are equal<\/li>\r\n \t
      • Direction vectors parallel AND lines share no points<\/strong> \u2192 Lines are parallel but distinct<\/li>\r\n \t
      • Direction vectors not parallel AND lines share a point<\/strong> \u2192 Lines intersect<\/li>\r\n \t
      • Direction vectors not parallel AND lines share no points<\/strong> \u2192 Lines are skew<\/li>\r\n<\/ul>\r\n<\/section>
        In 2D, if two lines aren't parallel, they must intersect. In 3D, they might \"miss\" each other entirely\u2014imagine one line running east-west on the ground floor of a building while another runs north-south on the second floor.<\/section><\/div>\r\n
        \r\n

        For each pair of lines, determine whether the lines are equal, parallel but not equal, skew, or intersecting.<\/p>\r\n\r\n

          \r\n \t
        1. [latex]L_1[\/latex]:\u00a0[latex]x=2s-1[\/latex],\u00a0[latex]y=s-1[\/latex],\u00a0[latex]z=s-4[\/latex]\r\n<\/span>[latex]L_2[\/latex]:\u00a0[latex]x=t-3[\/latex],\u00a0[latex]y=3t+8[\/latex],\u00a0[latex]z=5-2t[\/latex]<\/li>\r\n \t
        2. [latex]L_1[\/latex]:\u00a0[latex]x=-y=z[\/latex]\r\n<\/span>[latex]L_2[\/latex]:\u00a0[latex]\\frac{x-3}2=y=z-2[\/latex]<\/li>\r\n \t
        3. [latex]L_1[\/latex]:\u00a0[latex]x=6s-1[\/latex],\u00a0[latex]y=-2s[\/latex], [latex]z=3s+1[\/latex]\r\n<\/span>[latex]L_2[\/latex]:\u00a0[latex]\\frac{x-4}6=\\frac{y+3}{-2}=\\frac{z-1}3[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"849520983\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"849520983\"]\r\n
            \r\n \t
          1. Line [latex]L_1[\/latex] has direction vector [latex]{\\bf{v_1}}=\\langle2,1,1\\rangle[\/latex]; line [latex]L_2[\/latex] has direction vector [latex]{\\bf{v_2}}=\\langle1,3,-2\\rangle[\/latex]. Because the direction vectors are not parallel vectors, the lines are either intersecting or skew. To determine whether the lines intersect, we see if there is a point, [latex](x, y, z)[\/latex], that lies on both lines. To find this point, we use the parametric equations to create a system of equalities:\r\n<\/span>\r\n

            [latex]2s-1=t-3; \\quad s-1=3t+8; \\quad s-4=5-2t[\/latex].<\/p>\r\n\r\n<\/span>By the first equation, [latex]t=2s+2[\/latex]. Substituting into the second equation yields\r\n<\/span>\r\n

            [latex]\\begin{aligned}\r\ns-1&=3(2s+2)+8 \\\\\r\ns-1&=6s+6+8 \\\\\r\n5s&=-15 \\\\\r\ns&=-3.\r\n\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/span>Substitution into the third equation, however, yields a contradiction:\r\n<\/span>\r\n

            [latex]\\begin{aligned}\r\ns-4&=5-2(2s+2) \\\\\r\ns-4&=5-4s-4 \\\\\r\n5s&=5 \\\\\r\ns&=1.\r\n\\end{aligned}[\/latex]<\/p>\r\n\r\n<\/span>There is no single point that satisfies the parametric equations for [latex]L_1[\/latex] and [latex]L_2[\/latex] simultaneously. These lines do not intersect, so they are skew (see the following figure).\r\n<\/span>\r\n\r\n[caption id=\"attachment_5168\" align=\"aligncenter\" width=\"376\"]\"This Figure 7. Lines [latex]L_1[\/latex] and [latex]L_2[\/latex] are skew.[\/caption]<\/li>\r\n \t

          2. Line [latex]L_1[\/latex] has direction vector [latex]{\\bf{v_1}}=\\langle1,-1,1\\rangle[\/latex] and passes through the origin, [latex](0, 0, 0)[\/latex]. Line [latex]L_2[\/latex] has a different direction vector, [latex]{\\bf{v_2}}=\\langle2,1,1\\rangle[\/latex], so these lines are not parallel or equal. Let [latex]r[\/latex] represent the parameter for line [latex]L_1[\/latex] and let [latex]s[\/latex] represent the parameter for [latex]L_2[\/latex]:\r\n<\/span>\r\n

            [latex]\\begin{aligned}\r\nx&=r &\\quad x&=2s+3 \\\\\r\ny&=-r &\\quad y&=s \\\\\r\nz&=r &\\quad z&=s+2\r\n\\end{aligned}[\/latex].<\/p>\r\n\r\n<\/span>Solve the system of equations to find [latex]r=1[\/latex] and [latex]s=-1[\/latex]. If we need to find the point of intersection, we can substitute these parameters into the original equations to get [latex](1, -1, 1)[\/latex] (see the following figure).\r\n<\/span>\r\n\r\n[caption id=\"attachment_5169\" align=\"aligncenter\" width=\"514\"]\"This Figure 8. Lines [latex]L_1[\/latex] and [latex]L_2[\/latex] intersect at [latex](1, -1, 1)[\/latex].[\/caption]<\/li>\r\n \t

          3. Lines [latex]L_1[\/latex] and [latex]L_2[\/latex] have equivalent direction vectors: [latex]{\\bf{v}}=\\langle6,-2,3\\rangle[\/latex]. These two lines are parallel (see the following figure).\r\n<\/span>[caption id=\"attachment_5170\" align=\"aligncenter\" width=\"397\"]\"This Figure 9. Lines [latex]L_1[\/latex] and [latex]L_2[\/latex] are parallel.[\/caption]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"

            Relationships between Lines<\/h2>\n
            \n

            In two dimensions, two lines have only three possible relationships: they are equal (the same line), parallel but distinct, or they intersect at exactly one point. Three dimensions introduces a fourth possibility that doesn’t exist in 2D geometry.<\/p>\n

            When working with two lines in three-dimensional space, exactly one of these four relationships must be true:<\/p>\n

              \n
            1. Equal lines<\/strong>: The lines are identical (same direction, same points)<\/li>\n
            2. Parallel lines<\/strong>: The lines have the same direction but never meet<\/li>\n
            3. Intersecting lines<\/strong>: The lines cross at exactly one point<\/li>\n
            4. Skew lines<\/strong>: The lines are not parallel and do not intersect<\/li>\n<\/ol>\n
              \n

              skew lines<\/h3>\n

              Two lines are skew if they are not parallel but do not intersect. This relationship is unique to three-dimensional space and cannot occur in a plane.<\/p>\n<\/section>\n

              \"This
              Figure 5. In three dimensions, it is possible that two lines do not cross, even when they have different directions.<\/figcaption><\/figure>\n<\/div>\n
              \n

              To classify the relationship between two lines, we need to answer two key questions:<\/p>\n

                \n
              1. Are the direction vectors parallel?<\/strong> (Do the lines have the same or opposite directions?)<\/li>\n
              2. Do the lines share any points?<\/strong> (Is there a common point on both lines?)<\/li>\n<\/ol>\n

                The answers to these questions determine the relationship.<\/p>\n<\/div>\n

                <\/div>\n
                \n
                \"This
                Figure 6. Determine the relationship between two lines based on whether their direction vectors are parallel and whether they share a point.<\/figcaption><\/figure>\n
                \n

                line classification<\/h3>\n
                  \n
                • Direction vectors parallel AND lines share a point<\/strong> \u2192 Lines are equal<\/li>\n
                • Direction vectors parallel AND lines share no points<\/strong> \u2192 Lines are parallel but distinct<\/li>\n
                • Direction vectors not parallel AND lines share a point<\/strong> \u2192 Lines intersect<\/li>\n
                • Direction vectors not parallel AND lines share no points<\/strong> \u2192 Lines are skew<\/li>\n<\/ul>\n<\/section>\n
                  In 2D, if two lines aren’t parallel, they must intersect. In 3D, they might “miss” each other entirely\u2014imagine one line running east-west on the ground floor of a building while another runs north-south on the second floor.<\/section>\n<\/div>\n
                  \n

                  For each pair of lines, determine whether the lines are equal, parallel but not equal, skew, or intersecting.<\/p>\n

                    \n
                  1. [latex]L_1[\/latex]:\u00a0[latex]x=2s-1[\/latex],\u00a0[latex]y=s-1[\/latex],\u00a0[latex]z=s-4[\/latex]
                    \n<\/span>[latex]L_2[\/latex]:\u00a0[latex]x=t-3[\/latex],\u00a0[latex]y=3t+8[\/latex],\u00a0[latex]z=5-2t[\/latex]<\/li>\n
                  2. [latex]L_1[\/latex]:\u00a0[latex]x=-y=z[\/latex]
                    \n<\/span>[latex]L_2[\/latex]:\u00a0[latex]\\frac{x-3}2=y=z-2[\/latex]<\/li>\n
                  3. [latex]L_1[\/latex]:\u00a0[latex]x=6s-1[\/latex],\u00a0[latex]y=-2s[\/latex], [latex]z=3s+1[\/latex]
                    \n<\/span>[latex]L_2[\/latex]:\u00a0[latex]\\frac{x-4}6=\\frac{y+3}{-2}=\\frac{z-1}3[\/latex]<\/li>\n<\/ol>\n