{"id":2323,"date":"2025-08-12T21:45:10","date_gmt":"2025-08-12T21:45:10","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2323"},"modified":"2025-10-21T21:51:11","modified_gmt":"2025-10-21T21:51:11","slug":"operations-with-vectors-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/operations-with-vectors-learn-it-3\/","title":{"raw":"Operations with Vectors: Learn It 3","rendered":"Operations with Vectors: Learn It 3"},"content":{"raw":"

Finding the Dot Product of Two Vectors<\/h2>\r\nAs we discussed earlier in the section, scalar multiplication involves multiplying a vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the dot product<\/em> and the cross product<\/em>. We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses.\r\n\r\nThe dot product of two vectors involves multiplying two vectors together, and the result is a scalar.\r\n\r\n
\r\n

dot product<\/h3>\r\nThe dot product<\/strong> of two vectors [latex]\\boldsymbol{v}=\\langle a,b\\rangle [\/latex] and [latex]\\boldsymbol{v}=\\langle c,d\\rangle [\/latex] is the sum of the product of the horizontal components and the product of the vertical components.\r\n

[latex]\\boldsymbol{v}\\cdot \\boldsymbol{u}=ac+bd[\/latex]<\/p>\r\nTo find the angle between the two vectors, use the formula below.\r\n

[latex]\\cos \\theta =\\dfrac{\\boldsymbol{v}}{|\\boldsymbol{v}|}\\cdot \\dfrac{\\boldsymbol{u}}{|\\boldsymbol{u}|}[\/latex]<\/p>\r\n\r\n<\/section>

Find the dot product of [latex]\\boldsymbol{v}=\\langle 5,12\\rangle [\/latex] and [latex]\\boldsymbol{u}=\\langle -3,4\\rangle [\/latex].[reveal-answer q=\"734442\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"734442\"]Using the formula, we have\r\n

[latex]\\begin{align}\\boldsymbol{v}\\cdot \\boldsymbol{u}&=\\langle 5,12\\rangle \\cdot \\langle -3,4\\rangle \\\\ &=5\\cdot \\left(-3\\right)+12\\cdot 4 \\\\ &=-15+48 \\\\ &=33 \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

Find the dot product of v<\/em><\/strong>1<\/sub> = 5i<\/em><\/strong> + 2j<\/em><\/strong> and v<\/em><\/strong>2<\/sub> = 3i<\/em><\/strong> + 7j<\/em><\/strong>. Then, find the angle between the two vectors.[reveal-answer q=\"817790\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"817790\"]Finding the dot product, we multiply corresponding components.\r\n

[latex]\\begin{align}{\\boldsymbol{v}}_{1}\\cdot {\\boldsymbol{v}}_{2}&=\\langle 5,2\\rangle \\cdot \\langle 3,7\\rangle \\\\ &=5\\cdot 3+2\\cdot 7 \\\\ &=15+14 \\\\ &=29 \\end{align}[\/latex]<\/p>\r\nTo find the angle between them, we use the formula [latex]\\cos \\theta =\\dfrac{\\boldsymbol{v}}{|\\boldsymbol{v}|}\\cdot \\dfrac{\\boldsymbol{u}}{|\\boldsymbol{u}|}[\/latex].\r\n

[latex]\\begin{align}\\frac{\\boldsymbol{v}}{|\\boldsymbol{v}|}\\cdot \\frac{\\boldsymbol{u}}{|\\boldsymbol{u}|}&=\\langle \\frac{5}{\\sqrt{29}}+\\frac{2}{\\sqrt{29}}\\rangle \\cdot \\langle \\frac{3}{\\sqrt{58}}+\\frac{7}{\\sqrt{58}}\\rangle \\\\ &=\\frac{5}{\\sqrt{29}}\\cdot \\frac{3}{\\sqrt{58}}+\\frac{2}{\\sqrt{29}}\\cdot \\frac{7}{\\sqrt{58}} \\\\ &=\\frac{15}{\\sqrt{1682}}+\\frac{14}{\\sqrt{1682}}=\\frac{29}{\\sqrt{1682}} \\\\ &=0.707107 \\end{align}[\/latex]<\/p>\r\n

[latex]\\theta={\\cos }^{-1}\\left(0.707107\\right)=45^\\circ[\/latex]<\/p>\r\n\r\n

<\/div>\r\n\"Plot\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
Find the angle between [latex]\\boldsymbol{u}=\\langle -3,4\\rangle [\/latex] and [latex]\\boldsymbol{v}=\\langle 5,12\\rangle [\/latex].[reveal-answer q=\"953383\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"953383\"]Using the formula, [latex]\\theta ={\\cos }^{-1}\\left(\\dfrac{\\boldsymbol{u}}{|\\boldsymbol{u}|}\\cdot \\dfrac{\\boldsymbol{v}}{|\\boldsymbol{v}|}\\right)[\/latex],\r\n

[latex]\\begin{align} \\left(\\frac{\\boldsymbol{u}}{|\\boldsymbol{u}|}\\cdot \\frac{\\boldsymbol{v}}{|\\boldsymbol{v}|}\\right)&=\\frac{-3\\boldsymbol{i}+4\\boldsymbol{j}}{5}\\cdot \\frac{5\\boldsymbol{i}+12\\boldsymbol{j}}{13} \\\\ &=\\left(-\\frac{3}{5}\\cdot \\frac{5}{13}\\right)+\\left(\\frac{4}{5}\\cdot \\frac{12}{13}\\right) \\\\ &=-\\frac{15}{65}+\\frac{48}{65} \\\\ &=\\frac{33}{65}\\end{align}[\/latex]<\/p>\r\n

[latex]\\theta ={\\cos }^{-1}\\left(\\frac{33}{65}\\right) ={59.5}^{\\circ }[\/latex]<\/p>\r\n

\"Plot<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"

Finding the Dot Product of Two Vectors<\/h2>\n

As we discussed earlier in the section, scalar multiplication involves multiplying a vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the dot product<\/em> and the cross product<\/em>. We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses.<\/p>\n

The dot product of two vectors involves multiplying two vectors together, and the result is a scalar.<\/p>\n

\n

dot product<\/h3>\n

The dot product<\/strong> of two vectors [latex]\\boldsymbol{v}=\\langle a,b\\rangle[\/latex] and [latex]\\boldsymbol{v}=\\langle c,d\\rangle[\/latex] is the sum of the product of the horizontal components and the product of the vertical components.<\/p>\n

[latex]\\boldsymbol{v}\\cdot \\boldsymbol{u}=ac+bd[\/latex]<\/p>\n

To find the angle between the two vectors, use the formula below.<\/p>\n

[latex]\\cos \\theta =\\dfrac{\\boldsymbol{v}}{|\\boldsymbol{v}|}\\cdot \\dfrac{\\boldsymbol{u}}{|\\boldsymbol{u}|}[\/latex]<\/p>\n<\/section>\n

Find the dot product of [latex]\\boldsymbol{v}=\\langle 5,12\\rangle[\/latex] and [latex]\\boldsymbol{u}=\\langle -3,4\\rangle[\/latex].<\/p>\n