Time derivatives of unit vectors are always zero since unit vectors’ lengths are constant, regardless of coordinate system. True False<\/p>\n\n\n\n
The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n Correct Answer: False<\/strong><\/p>\n\n\n\n The statement \u201cTime derivatives of unit vectors are always zero since unit vectors\u2019 lengths are constant, regardless of coordinate system\u201d<\/strong> is false<\/strong> because it confuses the constancy of a vector\u2019s magnitude<\/em> with the constancy of its direction<\/em>.<\/p>\n\n\n\n While a unit vector<\/strong> indeed has a constant magnitude<\/strong> of 1, its direction can change over time<\/strong>, especially in non-Cartesian (curvilinear) coordinate systems<\/strong>, such as polar, cylindrical, or spherical coordinates. The time derivative of a vector accounts for changes in both magnitude and direction<\/strong>.<\/p>\n\n\n\n In Cartesian coordinates<\/strong>, the standard unit vectors i^,j^,k^\\hat{i}, \\hat{j}, \\hat{k} (pointing in the x, y, and z directions) are constant in both magnitude and direction. Thus, their time derivatives are indeed zero: di^dt=dj^dt=dk^dt=0\\frac{d\\hat{i}}{dt} = \\frac{d\\hat{j}}{dt} = \\frac{d\\hat{k}}{dt} = 0<\/p>\n\n\n\n However, in polar coordinates<\/strong>, the unit vectors r^\\hat{r} and \u03b8^\\hat{\\theta} change direction as the point moves. Their magnitudes remain 1, but the directions vary with the angular position \u03b8(t)\\theta(t). The time derivatives of these unit vectors are not zero<\/strong>: dr^dt=\u03b8\u02d9\u03b8^,d\u03b8^dt=\u2212\u03b8\u02d9r^\\frac{d\\hat{r}}{dt} = \\dot{\\theta} \\hat{\\theta}, \\quad \\frac{d\\hat{\\theta}}{dt} = -\\dot{\\theta} \\hat{r}<\/p>\n\n\n\n This shows that the directional change<\/strong> leads to a non-zero time derivative<\/strong> of the unit vector, even though its length stays constant<\/strong>.<\/p>\n\n\n\n Hence, it is incorrect to assume that the time derivative of a unit vector is always zero just because its magnitude doesn’t change. The key is whether the vector’s direction changes with time<\/strong>, which it often does in curvilinear coordinate systems.<\/p>\n\n\n\n Conclusion:<\/strong> The time derivative of a unit vector is not always zero. It depends on the coordinate system and whether the vector\u2019s direction changes over time.<\/p>\n","protected":false},"excerpt":{"rendered":" Time derivatives of unit vectors are always zero since unit vectors’ lengths are constant, regardless of coordinate system. True False The correct answer and explanation is: Correct Answer: False Explanation (Approx. 300 words): The statement \u201cTime derivatives of unit vectors are always zero since unit vectors\u2019 lengths are constant, regardless of coordinate system\u201d is false […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-19187","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/19187","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=19187"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/19187\/revisions"}],"predecessor-version":[{"id":19188,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/19187\/revisions\/19188"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=19187"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=19187"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=19187"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Explanation (Approx. 300 words):<\/h3>\n\n\n\n