{"id":5447,"date":"2022-06-02T18:23:59","date_gmt":"2022-06-02T18:23:59","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&p=5447"},"modified":"2022-10-21T00:13:52","modified_gmt":"2022-10-21T00:13:52","slug":"the-dot-product","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/the-dot-product\/","title":{"raw":"The Dot Product","rendered":"The Dot Product"},"content":{"raw":"
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Learning Objectives<\/h3>\r\n
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  • Calculate the dot product of two given vectors.<\/span><\/li>\r\n \t
  • Determine whether two given vectors are perpendicular.<\/span><\/li>\r\n \t
  • Find the direction cosines of a given vector.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n
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    The Dot Product and Its Properties<\/h2>\r\n

    We have already learned how to add and subtract vectors. In this chapter, we investigate two types of vector multiplication. The first type of vector multiplication is called the dot product, based on the notation we use for it, and it is defined as follows:<\/p>\r\n\r\n<\/section><\/div>\r\n

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    definition<\/h3>\r\n\r\n
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    The\u00a0dot product<\/span><\/strong>\u00a0of vectors [latex]\\mathbf{u} = \\langle u_1, u_2, u_3 \\rangle[\/latex] and [latex]\\mathbf{v} =\\langle v_1, v_2, v_3 \\rangle[\/latex] is given by the sum of the products of the components<\/p>\r\n\r\n

    [latex]\\mathbf{u \\cdot v}[\/latex][latex] = u_1v_1 + u_2v_2 + u_3v_3 [\/latex].<\/center><\/div>\r\n

    Note that if [latex]\\mathbf u[\/latex] and [latex]\\mathbf v[\/latex] are two-dimensional vectors, we calculate the dot product in a similar fashion. Thus, if [latex]\\mathbf{u} = \\langle u_1, u_2\\rangle[\/latex] and [latex]\\mathbf{v} =\\langle v_1, v_2 \\rangle[\/latex], then<\/p>\r\n\r\n

    [latex]\\mathbf{u \\cdot v}[\/latex][latex] = u_1v_1 + u_2v_2 [\/latex].<\/center>When two vectors are combined under addition or subtraction, the result is a vector. When two vectors are combined using the dot product, the result is a scalar. For this reason, the dot product is often called the\u00a0scalar product<\/em>. It may also be called the\u00a0inner product<\/em>.\r\n
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    Example: calculating dot products<\/h3>\r\n
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    1. Find the dot product of [latex]\\mathbf{u} = \\langle 3,5,2 \\rangle[\/latex] and\u00a0[latex]\\mathbf{v} =\\langle -1,3,0 \\rangle[\/latex].<\/li>\r\n \t
    2. Find the scalar product of [latex]\\mathbf{p} = 10\\mathbf{i} - 4\\mathbf{j} +7 \\mathbf{k}[\/latex] and\u00a0[latex]\\mathbf{q} = -2\\mathbf{i} + \\mathbf{j} + 6\\mathbf{k}[\/latex].<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"471238733\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"471238733\"]\r\n
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      1. Substitute the vector components into the formula for the dot product:\r\n<\/span>\r\n
        \r\n[latex]\r\n\\begin{align*}\r\n\\mathbf{u \\cdot v} &= u_1v_1 + u_2v_2 + u_3v_3 \\\\\r\n&= 3(-1)+5(3)+2(0)= -3+15+0 = 12.\\\\\r\n\\end{align*}\r\n[\/latex]<\/center><\/div><\/li>\r\n \t
      2. The calculation is the same if the vectors are written using standard unit vectors. We still have three components for each vector to substitute into the formula for the dot product:\r\n
        \r\n[latex]\r\n\\begin{align*}\r\n\\mathbf{p \\cdot q} &= u_1v_1 + u_2v_2 + u_3v_3 \\\\\r\n&= 10(-2)+(-4)(1)+7(6)= -20-4+42 = 18.\\\\\r\n\\end{align*}\r\n[\/latex]<\/center><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n
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        try it<\/h3>\r\nFind [latex]\\mathbf{u \\cdot v}[\/latex], where [latex]\\mathbf{u} = \\langle 2,9,-1 \\rangle[\/latex] and [latex]\\mathbf{v} =\\langle -3,1,-4 \\rangle[\/latex].\r\n\r\n[reveal-answer q=\"847271036\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"847271036\"]\r\n\r\n[latex]7[\/latex]<\/span>\r\n\r\n[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\nWatch the following video to see the worked solution to the above Try IT.\r\n\r\n