[amp_mcq option1=”30″ option2=”60″ option3=”90″ option4=”120″ correct=”option3″]<\/p>\n
The correct answer is $\\boxed{\\text{C}}$.<\/p>\n
The angle between two vectors $\\overrightarrow{\\rm{a}}$ and $\\overrightarrow{\\rm{b}}$ is given by the following formula:<\/p>\n
$$\\cos \\theta = \\frac{\\overrightarrow{\\rm{a}} \\cdot \\overrightarrow{\\rm{b}}}{\\left\\| \\overrightarrow{\\rm{a}} \\right\\| \\left\\| \\overrightarrow{\\rm{b}} \\right\\|}$$<\/p>\n
where $\\theta$ is the angle between the two vectors, $\\overrightarrow{\\rm{a}} \\cdot \\overrightarrow{\\rm{b}}$ is the dot product of the two vectors, and $\\left\\| \\overrightarrow{\\rm{a}} \\right\\|$ and $\\left\\| \\overrightarrow{\\rm{b}} \\right\\|$ are the norms of the two vectors.<\/p>\n
In this case, we have:<\/p>\n
$$\\overrightarrow{\\rm{a}} = \\frac{{\\sqrt 3 }}{2}{\\rm{\\hat i}} + \\frac{1}{2}{\\rm{\\hat j}}$$<\/p>\n
$$\\overrightarrow{\\rm{b}} = \\frac{{ – \\sqrt 3 }}{2}{\\rm{\\hat i}} + \\frac{1}{2}{\\rm{\\hat j}}$$<\/p>\n
Therefore, we have:<\/p>\n
$$\\overrightarrow{\\rm{a}} \\cdot \\overrightarrow{\\rm{b}} = \\frac{{\\sqrt 3 }}{2} \\cdot \\frac{{ – \\sqrt 3 }}{2} + \\frac{1}{2} \\cdot \\frac{1}{2} = -\\frac{3}{4}$$<\/p>\n
$$\\left\\| \\overrightarrow{\\rm{a}} \\right\\| = \\sqrt{\\left(\\frac{{\\sqrt 3 }}{2}\\right)^2 + \\left(\\frac{1}{2}\\right)^2} = \\frac{\\sqrt{3}}{2}$$<\/p>\n
$$\\left\\| \\overrightarrow{\\rm{b}} \\right\\| = \\sqrt{\\left(\\frac{{ – \\sqrt 3 }}{2}\\right)^2 + \\left(\\frac{1}{2}\\right)^2} = \\frac{\\sqrt{3}}{2}$$<\/p>\n
Therefore, we have:<\/p>\n
$$\\cos \\theta = \\frac{\\overrightarrow{\\rm{a}} \\cdot \\overrightarrow{\\rm{b}}}{\\left\\| \\overrightarrow{\\rm{a}} \\right\\| \\left\\| \\overrightarrow{\\rm{b}} \\right\\|} = \\frac{-\\frac{3}{4}}{\\frac{\\sqrt{3}}{2} \\cdot \\frac{\\sqrt{3}}{2}} = -\\frac{3}{3} = -1$$<\/p>\n
The angle between two vectors is $\\boxed{90^\\circ}$ when the dot product of the two vectors is equal to $-1$.<\/p>\n","protected":false},"excerpt":{"rendered":"
[amp_mcq option1=”30″ option2=”60″ option3=”90″ option4=”120″ correct=”option3″]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[690],"tags":[],"class_list":["post-20192","post","type-post","status-publish","format-standard","hentry","category-calculus","no-featured-image-padding"],"yoast_head":"\n