{"id":860,"date":"2025-10-14T14:50:29","date_gmt":"2025-10-14T14:50:29","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&p=860"},"modified":"2026-05-26T14:25:18","modified_gmt":"2026-05-26T14:25:18","slug":"vector-basics-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/vector-basics-learn-it-2\/","title":{"raw":"Vector Basics: Learn It 2","rendered":"Vector Basics: Learn It 2"},"content":{"raw":"

Combining Vectors<\/h2>\r\n

Vectors appear everywhere in real-world situations involving forces, velocities, and other directional quantities. Understanding how to combine and manipulate vectors is essential for solving these problems.<\/p>\r\n

Consider a boat crossing a river. The boat's motor creates a force in one direction, while the river current creates a force in another direction. To predict where the boat will actually go, you need to account for both forces\u2014their magnitudes and directions.<\/p>\r\nA second example that involves vectors is a quarterback throwing a football. The quarterback does not throw the ball parallel to the ground; instead, he aims up into the air. The velocity of his throw can be represented by a vector. If we know how hard he throws the ball (magnitude\u2014in this case, speed), and the angle (direction), we can tell how far the ball will travel down the field.\r\n\r\nBefore working with vector operations, let's clarify an important distinction. A scalar<\/strong> is a real number that has only magnitude\u2014like temperature, mass, or speed. Unlike vectors, scalars have no direction component.\r\n\r\n

\r\n

Scalar vs. Vector<\/strong>:<\/p>\r\n\r\n

    \r\n \t
  • Scalar<\/strong>: Has magnitude only (examples: [latex]5[\/latex], [latex]-3.2[\/latex], [latex]\\pi[\/latex])<\/li>\r\n \t
  • Vector<\/strong>: Has both magnitude and direction (examples: velocity, force, displacement)<\/li>\r\n<\/ul>\r\n<\/section>\r\n

    Scalar Multiplication<\/h3>\r\n

    When you multiply a vector by a scalar, you change the vector's magnitude but affect its direction in a predictable way. This operation is called scalar multiplication<\/strong>.<\/p>\r\n\r\n

    \r\n

    scalar multiplication<\/h3>\r\n

    For a vector [latex]\\mathbf{v}[\/latex] and scalar [latex]k[\/latex], the product [latex]k\\mathbf{v}[\/latex] is a vector where:<\/p>\r\n\r\n

      \r\n \t
    • Magnitude<\/strong>: [latex]|k|[\/latex] times the magnitude of [latex]\\mathbf{v}[\/latex]<\/li>\r\n \t
    • Direction<\/strong>: Same as [latex]\\mathbf{v}[\/latex] if [latex]k > 0[\/latex], opposite if [latex]k < 0[\/latex]<\/li>\r\n \t
    • If [latex]k = 0[\/latex] or [latex]\\mathbf{v} = \\mathbf{0}[\/latex], then [latex]k\\mathbf{v} = \\mathbf{0}[\/latex]<\/li>\r\n<\/ul>\r\n<\/section>
      Think of scalar multiplication as stretching or shrinking a vector. If [latex]k = 2[\/latex], you double the vector's length. If [latex]k = 0.5[\/latex], you halve it. If [latex]k = -1[\/latex], you flip the vector to point in the opposite direction without changing its length.<\/section>\r\n

      When [latex]k = -1[\/latex], we write the result as:<\/p>\r\n

      [latex]k\\mathbf{v} = (-1)\\mathbf{v} = -\\mathbf{v}[\/latex]<\/p>\r\n

      The vector [latex]-\\mathbf{v}[\/latex] has the same magnitude as [latex]\\mathbf{v}[\/latex] but points in the opposite direction.<\/p>\r\n\r\n[caption id=\"attachment_678\" align=\"aligncenter\" width=\"435\"]\"This Figure 4. (a) The original vector [latex]{\\bf{v}}[\/latex] has length [latex]n[\/latex] units. (b) The length of [latex]2{\\bf{v}}[\/latex] equals [latex]2n[\/latex] units. (c) The length of\u00a0[latex]{\\bf{v}}{\/}2[\/latex] is [latex]n{\/}2[\/latex] units. (d) The vectors [latex]{\\bf{v}}[\/latex] and\u00a0[latex]-{\\bf{v}}[\/latex] have the same length but opposite directions.[\/caption]

      \r\n

      Wind blowing north at [latex]15[\/latex] mph can be represented by vector [latex]\\mathbf{v}[\/latex]. Then:<\/p>\r\n\r\n

        \r\n \t
      • [latex]2\\mathbf{v}[\/latex] represents wind blowing north at [latex]30[\/latex] mph<\/li>\r\n \t
      • [latex]0.5\\mathbf{v}[\/latex] represents wind blowing north at [latex]7.5[\/latex] mph<\/li>\r\n \t
      • [latex]-\\mathbf{v}[\/latex] represents wind blowing south at [latex]15[\/latex] mph<\/li>\r\n<\/ul>\r\n<\/section>\r\n

        Vector Addition<\/h3>\r\nAdding vectors is different from adding regular numbers because each vector has its own direction. You need graphical methods to combine them properly.\r\n

        Triangle Method<\/h4>\r\n

        The most common approach places the initial point of the second vector at the terminal point of the first vector.<\/p>\r\n

        Think about this in terms of displacement. If you walk from point [latex]A[\/latex] to point [latex]B[\/latex] (following vector [latex]\\mathbf{v}[\/latex]), then walk from point [latex]B[\/latex] to point [latex]C[\/latex] (following vector [latex]\\mathbf{w}[\/latex]), your total displacement is the same as walking directly from [latex]A[\/latex] to [latex]C[\/latex]. This direct path represents the vector sum [latex]\\mathbf{v} + \\mathbf{w}[\/latex].<\/p>\r\n

        The order doesn't matter\u2014you get the same result whether you add [latex]\\mathbf{v} + \\mathbf{w}[\/latex] or [latex]\\mathbf{w} + \\mathbf{v}[\/latex]. This is called the commutative property<\/strong> of vector addition.<\/p>\r\n

        Notice that the three vectors [latex]\\mathbf{v}[\/latex], [latex]\\mathbf{w}[\/latex], and [latex]\\mathbf{v} + \\mathbf{w}[\/latex] form a triangle. From basic geometry, the length of any side of a triangle must be less than or equal to the sum of the other two sides. This gives us the triangle inequality<\/strong>:<\/p>\r\n

        [latex]||\\mathbf{v} + \\mathbf{w}|| \\leq ||\\mathbf{v}|| + ||\\mathbf{w}||[\/latex]<\/p>\r\n

        The only time we get equality is when [latex]\\mathbf{v}[\/latex] and [latex]\\mathbf{w}[\/latex] point in exactly the same direction.<\/p>\r\n\r\n

        Parallelogram Method<\/h4>\r\n

        An alternative approach places both vectors at the same initial point, then draws a parallelogram using the vectors as adjacent sides. The diagonal of the parallelogram represents the vector sum.<\/p>\r\n

        Both methods give identical results\u2014use whichever seems clearer for your problem.<\/p>\r\n\r\n[caption id=\"attachment_680\" align=\"aligncenter\" width=\"409\"]\"This Figure 5. (a) When adding vectors by the triangle method, the initial point of [latex]{\\bf{w}}[\/latex] is the terminal point of [latex]{\\bf{v}}[\/latex]. (b) When adding vectors by the parallelogram method, the vectors [latex]{\\bf{v}}[\/latex] and [latex]{\\bf{w}}[\/latex] have the same initial point.[\/caption]

        \r\n

        vector addition<\/h3>\r\nTo add vectors [latex]\\mathbf{v}[\/latex] and [latex]\\mathbf{w}[\/latex] graphically, place the initial point of [latex]\\mathbf{w}[\/latex] at the terminal point of [latex]\\mathbf{v}[\/latex]. The vector sum [latex]\\mathbf{v} + \\mathbf{w}[\/latex] runs from the initial point of [latex]\\mathbf{v}[\/latex] to the terminal point of [latex]\\mathbf{w}[\/latex].\r\n\r\n<\/section>\r\n

        Vector Subtraction<\/h3>\r\n

        Vector subtraction uses the same principle as adding the negative of a vector. We define:<\/p>\r\n

        [latex]\\mathbf{v} - \\mathbf{w} = \\mathbf{v} + (-\\mathbf{w})[\/latex]<\/p>\r\n

        Graphically, [latex]\\mathbf{v} - \\mathbf{w}[\/latex] is the vector that points from the terminal point of [latex]\\mathbf{w}[\/latex] to the terminal point of [latex]\\mathbf{v}[\/latex].<\/p>\r\n\r\n[caption id=\"attachment_681\" align=\"aligncenter\" width=\"487\"]\"This Figure 6. (a) The vector difference [latex]{\\bf{v-w}}[\/latex] is depicted by drawing a vector from the terminal point of [latex]{\\bf{w}}[\/latex] to the terminal point of\u00a0[latex]{\\bf{v}}[\/latex]. (b) The vector [latex]{\\bf{v-w}}[\/latex] is equivalent to the vector [latex]{\\bf{v+(-w)}}[\/latex].[\/caption]In Figure 4(a), the initial point of [latex]\\bf{v + w}[\/latex] is the initial point of [latex]\\bf{v}[\/latex]. The terminal point of [latex]\\bf{v + w}[\/latex] is the terminal point of [latex]\\bf{w}[\/latex] . These three vectors form the sides of a triangle. It follows that the length of any one side is less than the sum of the lengths of the remaining sides. So we have\r\n\r\n

        [latex]\\bf{|| v + w|| \\leq ||v|| + ||w||}[\/latex].<\/center>This is known more generally as the triangle inequality<\/strong>. There is one case, however, when the resultant vector [latex]\\bf{u + v}[\/latex] has the same magnitude as the sum of the magnitudes of [latex]\\bf{u}[\/latex] and [latex]\\bf{v}[\/latex]. This happens only when [latex]\\bf{u}[\/latex] and [latex]\\bf{v}[\/latex] have the same direction.\r\n\r\n
        Given the vectors [latex]\\bf{v}[\/latex] and [latex]\\bf{w}[\/latex] shown in Figure 6, sketch the vectors.\r\n
          \r\n \t
        1. [latex]\\bf{3w}[\/latex]<\/li>\r\n \t
        2. [latex]\\bf{v + w}[\/latex]<\/li>\r\n \t
        3. [latex]\\bf{2v - w}[\/latex]<\/li>\r\n<\/ol>\r\n[caption id=\"attachment_683\" align=\"aligncenter\" width=\"227\"]\"This Figure 7. Vectors\u00a0[latex]{\\bf{v}}[\/latex] and [latex]{\\bf{w}}[\/latex] lie in the same plane.[\/caption]\r\n
          <\/div>\r\n
          [reveal-answer q=\"fs-id1167794055165\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1167794055165\"]\r\n
            a. The vector [latex]\\bf{3w}[\/latex] has the same direction as [latex]\\bf{w}[\/latex]; it is three times as long as [latex]\\bf{w}[\/latex].<\/ol>\r\n[caption id=\"attachment_686\" align=\"aligncenter\" width=\"337\"]\"This Figure 8.[\/caption]\r\n\r\nVector [latex]\\bf{3w}[\/latex] has the same direction as [latex]\\bf{w}[\/latex] and is three times as long.\r\n
              b. Use either addition method to find [latex]\\bf{v + w}[\/latex].<\/ol>\r\n[caption id=\"attachment_687\" align=\"aligncenter\" width=\"486\"]\"This Figure 9. To find\u00a0[latex]{\\bf{v+w}}[\/latex], align the vectors at their initial points or place the initial point of one vector at the terminal point of the other. (a) The vector [latex]{\\bf{v+w}}[\/latex] is the diagonal of the parallelogram with sides [latex]{\\bf{v}}[\/latex] and [latex]{\\bf{w}}[\/latex] (b) The vector [latex]{\\bf{v+w}}[\/latex] is the third side of a triangle formed with [latex]{\\bf{w}}[\/latex] placed at the terminal point of [latex]{\\bf{v}}[\/latex].[\/caption]\r\n
                c. To find [latex]\\bf{2v - w}[\/latex], we can first rewrite the expression as [latex]\\bf{2v + (-w)}[\/latex]. Then we can draw the vector [latex]\\bf{-w}[\/latex], then add it to the vector [latex]\\bf{2v}[\/latex].<\/ol>\r\n[caption id=\"attachment_688\" align=\"aligncenter\" width=\"195\"]\"This Figure 10. To find [latex]2{\\bf{v-w}}[\/latex], simply add [latex]2{\\bf{v+(-w)}}[\/latex].[\/caption][\/hidden-answer]<\/div>\r\n<\/section>
                [ohm_question hide_question_numbers=1]325831[\/ohm_question]<\/section>","rendered":"

                Combining Vectors<\/h2>\n

                Vectors appear everywhere in real-world situations involving forces, velocities, and other directional quantities. Understanding how to combine and manipulate vectors is essential for solving these problems.<\/p>\n

                Consider a boat crossing a river. The boat’s motor creates a force in one direction, while the river current creates a force in another direction. To predict where the boat will actually go, you need to account for both forces\u2014their magnitudes and directions.<\/p>\n

                A second example that involves vectors is a quarterback throwing a football. The quarterback does not throw the ball parallel to the ground; instead, he aims up into the air. The velocity of his throw can be represented by a vector. If we know how hard he throws the ball (magnitude\u2014in this case, speed), and the angle (direction), we can tell how far the ball will travel down the field.<\/p>\n

                Before working with vector operations, let’s clarify an important distinction. A scalar<\/strong> is a real number that has only magnitude\u2014like temperature, mass, or speed. Unlike vectors, scalars have no direction component.<\/p>\n

                \n

                Scalar vs. Vector<\/strong>:<\/p>\n

                  \n
                • Scalar<\/strong>: Has magnitude only (examples: [latex]5[\/latex], [latex]-3.2[\/latex], [latex]\\pi[\/latex])<\/li>\n
                • Vector<\/strong>: Has both magnitude and direction (examples: velocity, force, displacement)<\/li>\n<\/ul>\n<\/section>\n

                  Scalar Multiplication<\/h3>\n

                  When you multiply a vector by a scalar, you change the vector’s magnitude but affect its direction in a predictable way. This operation is called scalar multiplication<\/strong>.<\/p>\n

                  \n

                  scalar multiplication<\/h3>\n

                  For a vector [latex]\\mathbf{v}[\/latex] and scalar [latex]k[\/latex], the product [latex]k\\mathbf{v}[\/latex] is a vector where:<\/p>\n

                    \n
                  • Magnitude<\/strong>: [latex]|k|[\/latex] times the magnitude of [latex]\\mathbf{v}[\/latex]<\/li>\n
                  • Direction<\/strong>: Same as [latex]\\mathbf{v}[\/latex] if [latex]k > 0[\/latex], opposite if [latex]k < 0[\/latex]<\/li>\n
                  • If [latex]k = 0[\/latex] or [latex]\\mathbf{v} = \\mathbf{0}[\/latex], then [latex]k\\mathbf{v} = \\mathbf{0}[\/latex]<\/li>\n<\/ul>\n<\/section>\n
                    Think of scalar multiplication as stretching or shrinking a vector. If [latex]k = 2[\/latex], you double the vector’s length. If [latex]k = 0.5[\/latex], you halve it. If [latex]k = -1[\/latex], you flip the vector to point in the opposite direction without changing its length.<\/section>\n

                    When [latex]k = -1[\/latex], we write the result as:<\/p>\n

                    [latex]k\\mathbf{v} = (-1)\\mathbf{v} = -\\mathbf{v}[\/latex]<\/p>\n

                    The vector [latex]-\\mathbf{v}[\/latex] has the same magnitude as [latex]\\mathbf{v}[\/latex] but points in the opposite direction.<\/p>\n

                    \"This
                    Figure 4. (a) The original vector [latex]{\\bf{v}}[\/latex] has length [latex]n[\/latex] units. (b) The length of [latex]2{\\bf{v}}[\/latex] equals [latex]2n[\/latex] units. (c) The length of\u00a0[latex]{\\bf{v}}{\/}2[\/latex] is [latex]n{\/}2[\/latex] units. (d) The vectors [latex]{\\bf{v}}[\/latex] and\u00a0[latex]-{\\bf{v}}[\/latex] have the same length but opposite directions.<\/figcaption><\/figure>\n
                    \n

                    Wind blowing north at [latex]15[\/latex] mph can be represented by vector [latex]\\mathbf{v}[\/latex]. Then:<\/p>\n

                      \n
                    • [latex]2\\mathbf{v}[\/latex] represents wind blowing north at [latex]30[\/latex] mph<\/li>\n
                    • [latex]0.5\\mathbf{v}[\/latex] represents wind blowing north at [latex]7.5[\/latex] mph<\/li>\n
                    • [latex]-\\mathbf{v}[\/latex] represents wind blowing south at [latex]15[\/latex] mph<\/li>\n<\/ul>\n<\/section>\n

                      Vector Addition<\/h3>\n

                      Adding vectors is different from adding regular numbers because each vector has its own direction. You need graphical methods to combine them properly.<\/p>\n

                      Triangle Method<\/h4>\n

                      The most common approach places the initial point of the second vector at the terminal point of the first vector.<\/p>\n

                      Think about this in terms of displacement. If you walk from point [latex]A[\/latex] to point [latex]B[\/latex] (following vector [latex]\\mathbf{v}[\/latex]), then walk from point [latex]B[\/latex] to point [latex]C[\/latex] (following vector [latex]\\mathbf{w}[\/latex]), your total displacement is the same as walking directly from [latex]A[\/latex] to [latex]C[\/latex]. This direct path represents the vector sum [latex]\\mathbf{v} + \\mathbf{w}[\/latex].<\/p>\n

                      The order doesn’t matter\u2014you get the same result whether you add [latex]\\mathbf{v} + \\mathbf{w}[\/latex] or [latex]\\mathbf{w} + \\mathbf{v}[\/latex]. This is called the commutative property<\/strong> of vector addition.<\/p>\n

                      Notice that the three vectors [latex]\\mathbf{v}[\/latex], [latex]\\mathbf{w}[\/latex], and [latex]\\mathbf{v} + \\mathbf{w}[\/latex] form a triangle. From basic geometry, the length of any side of a triangle must be less than or equal to the sum of the other two sides. This gives us the triangle inequality<\/strong>:<\/p>\n

                      [latex]||\\mathbf{v} + \\mathbf{w}|| \\leq ||\\mathbf{v}|| + ||\\mathbf{w}||[\/latex]<\/p>\n

                      The only time we get equality is when [latex]\\mathbf{v}[\/latex] and [latex]\\mathbf{w}[\/latex] point in exactly the same direction.<\/p>\n

                      Parallelogram Method<\/h4>\n

                      An alternative approach places both vectors at the same initial point, then draws a parallelogram using the vectors as adjacent sides. The diagonal of the parallelogram represents the vector sum.<\/p>\n

                      Both methods give identical results\u2014use whichever seems clearer for your problem.<\/p>\n

                      \"This
                      Figure 5. (a) When adding vectors by the triangle method, the initial point of [latex]{\\bf{w}}[\/latex] is the terminal point of [latex]{\\bf{v}}[\/latex]. (b) When adding vectors by the parallelogram method, the vectors [latex]{\\bf{v}}[\/latex] and [latex]{\\bf{w}}[\/latex] have the same initial point.<\/figcaption><\/figure>\n
                      \n

                      vector addition<\/h3>\n

                      To add vectors [latex]\\mathbf{v}[\/latex] and [latex]\\mathbf{w}[\/latex] graphically, place the initial point of [latex]\\mathbf{w}[\/latex] at the terminal point of [latex]\\mathbf{v}[\/latex]. The vector sum [latex]\\mathbf{v} + \\mathbf{w}[\/latex] runs from the initial point of [latex]\\mathbf{v}[\/latex] to the terminal point of [latex]\\mathbf{w}[\/latex].<\/p>\n<\/section>\n

                      Vector Subtraction<\/h3>\n

                      Vector subtraction uses the same principle as adding the negative of a vector. We define:<\/p>\n

                      [latex]\\mathbf{v} - \\mathbf{w} = \\mathbf{v} + (-\\mathbf{w})[\/latex]<\/p>\n

                      Graphically, [latex]\\mathbf{v} - \\mathbf{w}[\/latex] is the vector that points from the terminal point of [latex]\\mathbf{w}[\/latex] to the terminal point of [latex]\\mathbf{v}[\/latex].<\/p>\n

                      \"This
                      Figure 6. (a) The vector difference [latex]{\\bf{v-w}}[\/latex] is depicted by drawing a vector from the terminal point of [latex]{\\bf{w}}[\/latex] to the terminal point of\u00a0[latex]{\\bf{v}}[\/latex]. (b) The vector [latex]{\\bf{v-w}}[\/latex] is equivalent to the vector [latex]{\\bf{v+(-w)}}[\/latex].<\/figcaption><\/figure>\n

                      In Figure 4(a), the initial point of [latex]\\bf{v + w}[\/latex] is the initial point of [latex]\\bf{v}[\/latex]. The terminal point of [latex]\\bf{v + w}[\/latex] is the terminal point of [latex]\\bf{w}[\/latex] . These three vectors form the sides of a triangle. It follows that the length of any one side is less than the sum of the lengths of the remaining sides. So we have<\/p>\n

                      [latex]\\bf{|| v + w|| \\leq ||v|| + ||w||}[\/latex].<\/div>\n

                      This is known more generally as the triangle inequality<\/strong>. There is one case, however, when the resultant vector [latex]\\bf{u + v}[\/latex] has the same magnitude as the sum of the magnitudes of [latex]\\bf{u}[\/latex] and [latex]\\bf{v}[\/latex]. This happens only when [latex]\\bf{u}[\/latex] and [latex]\\bf{v}[\/latex] have the same direction.<\/p>\n

                      Given the vectors [latex]\\bf{v}[\/latex] and [latex]\\bf{w}[\/latex] shown in Figure 6, sketch the vectors.<\/p>\n
                        \n
                      1. [latex]\\bf{3w}[\/latex]<\/li>\n
                      2. [latex]\\bf{v + w}[\/latex]<\/li>\n
                      3. [latex]\\bf{2v - w}[\/latex]<\/li>\n<\/ol>\n
                        \"This
                        Figure 7. Vectors\u00a0[latex]{\\bf{v}}[\/latex] and [latex]{\\bf{w}}[\/latex] lie in the same plane.<\/figcaption><\/figure>\n
                        <\/div>\n
                        \n