In this video I go further into the properties of the cross product and this time derive the vector triple product. Note that this is Property 6 of Theorem 3 from my earlier video, and I solve it as part of Exercise 5. The vector triple product involves the cross product of a cross product and it produces a vector with a pair of dot products. This has applications in astronomy such as in Kepler's First Law of planetary motion, so it is important to learn how the Property is derived. This video was taken from my earlier video listed below: - Vectors and the Geometry of Space: The Cross Product: https://youtu.be/k8GRt95i-Gc - Video notes: https://peakd.com/hive-128780/@mes/vectors-and-the-geometry-of-space-the-cross-product - Playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FuRJ8rg-YVQvfPoPOwhuRW . ------------------------------------------------------ Become a MES Super Fan! https://www.youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate SUBSCRIBE via EMAIL: https://mes.fm/subscribe MES Links: https://mes.fm/links MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps ... https://www.youtube.com/watch?v=KKRMlFaNtes

In this video I go over another example on graphing parametric equations and this time look at the equations x = sin(t) and y = sin^2(t). The first thing to notice when looking at these parametric equations is that y = x^2 which is a simple parabola. But the domain and range of the parametric equations must also be taken into account when sketching the curve. Although the shape is a parabola we must look at the nature of the parametric equations to see how the path is traced along the curve. Since the parametric equations are trigonometric functions, they fluctuate periodically. Thus the path traced is a parabola but that moves back and forth along it an infinite amount of times. Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIht9Blk1bEEKchM9u0w View Video Notes on Steemit: https://steemit.com/mathematics/@mes/parametric-curves-example-4 Related Videos: Parametric Curves: Example 3: https://youtu.be/VpiJdEviBbk Parametric Curves: Example 2: https://youtu.be/SBkzZDlqTTE Parametric Curves: Example 1: https://youtu.be/0k7QyWa9-HA Parametric Equations and Curves: https://youtu.be/Kd3XF4LZoFE Parametric Equations and Polar Coordinates: https://youtu.be/usSors49Gdw Trigonometry: Sine, Cosine and Tan Functions: http://youtu.be/WKTIlF2oWw8 . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: Official Website: https://MES.fm Steemit: https://steemit.com/@mes Gab: https://gab.ai/matheasysolutions Minds: https://minds.com/matheasysolutions Twitter: https://twitter.com/MathEasySolns Facebook: https://fb.com/MathEasySolutions LinkedIn: https://mes.fm/linkedin Pinterest: https://pinterest.com/MathEasySolns Instagram: https://instagram.com/MathEasySolutions Email me: contact@mes.fm Try our Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Try our Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps ... https://www.youtube.com/watch?v=WZYIS1My-g4

In #MESExperiments 17 I demonstrate that even on a slanted needle a gyroscope can magically rise on its own. This demonstrates that a gyroscope's self-balancing mechanism does not require a vertical, symmetrical, or even centered base to precess about. The needle was tilted at 24° and the starting gyroscope tilt angle was 32°; note that both angles are measured from the vertical. The gyro, as on the vertical needle in Experiment 16, rises very quickly and this time rises in 23 seconds. This experiment also illustrates the magical behavior of gyroscopes to modify their “inertia”, effectively disappearing and reappearing their “mass”. This is shown by initially the gyroscope precesses with zero horizontal movement of the overall needle/clamp base but then magically regains this “mass” as the gyro speed slows down and spirals downwards while moving the entire base aggressively to the left and just before the gyro throws itself off. Magical stuff! The original unedited gyro video is shown here: https://youtu.be/2tkkiGtsnpI The sequence of events is listed below as well as the screenshots of the experiment. - Needle tilt angle from the vertical: 24° - Initial gyro angle: 32° - Time to rise: (0:28 - 0:05) = 23 seconds. - Sleep time: (1:06 - 0:28) = 38 seconds. - Time to fall: (1:18 - 0:05) = 1:13 = 73 seconds. The weight of the gyroscope is 100.29 grams and the needle width is about 0.5 mm. For screenshots of the experiment progression, make sure to check out the accompanying Hive post: https://peakd.com/mesexperiments/@mes/mesexperiments-17-gyroscopes-can-even-rise-on-a-slanted-needle Stay Tuned for #MESExperiments 18… Related Videos: ?#MESExperiments 16: Gyroscopes Can Even Rise on the Tip of a Thin Needle! #WOW: https://peakd.com/mesexperiments/@mes/mesexperiments-16-gyroscopes-can-even-rise-on-the-tip-of-a-thin-needle-wow #MESExperiments Video Series: https://peakd.com/mesexperiments/@mes/list DRAFT #MESExperiments Video Series: https://mes.fm/experiments-draft #AntiGravity Video Series: https://peakd.com/antigravity/@mes/series #FreeEnergy Video Series: https://mes.fm/freeenergy-playlist . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: MES Truth: https://mes.fm/truth Official Website: https://MES.fm Hive: https://peakd.com/@mes MORE Links: https://linktr.ee/matheasy Email me: contact@mes.fm Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps

In this video I go over another example on Integration by Parts as well as giving an introduction to integration by parts for definite integrals. In this example I solve the definite integral from x = 0 to x = 1 of the function arctan(x) or inverse tan(x). The integration by parts for definite integrals is very similar to the formula for indefinite integrals but each part needs to be evaluated using the 2nd part of the fundamental theorem of calculus. Download the notes in my video: http://1drv.ms/1KBZY7P View Video Notes on Steemit: https://steemit.com/mathematics/@mes/integration-by-parts-example-5-definite-integrals Related Videos: Integration by Parts: Example 4: http://youtu.be/xk_GKQPWbtI Integration by Parts: Example 3: Integrating Twice by Parts: http://youtu.be/Aq0dRUL4Wss Integration by Parts: Example 2: ln(x): http://youtu.be/wVjra6DCZzE Integration by Parts: Example 1: http://youtu.be/K8pReMFOJl4 Integration by Parts: Proof: http://youtu.be/TZhEOct5u_M Derivative Rules - Proof of the Product Rule: http://youtu.be/EIjvGJhDAOk The Substitution Rule for Integrals: http://youtu.be/VsLC-0g6hVg Derivative Rules: Proof of Chain Rule: http://youtu.be/tYDDpKzP-VU The Definite Integral - Brief Introduction: http://youtu.be/vhMP5SKbQjU Fundamental Theorem of Calculus - Intro and Proof of Part 2 of the Theorem: http://youtu.be/yuIl-BPQHss Fundamental Theorem of Calculus - Introduction and Part 1 of the Theorem: http://youtu.be/3o8Q6UJzJyk Fundamental Theorem of Calculus - Proof of Part 1 of the Theorem: http://youtu.be/CAqTwiPxYwU . ------------------------------------------------------ SUBSCRIBE via EMAIL: https://mes.fm/subscribe DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate Like, Subscribe, Favorite, and Comment Below! Follow us on: Official Website: https://MES.fm Steemit: https://steemit.com/@mes Gab: https://gab.ai/matheasysolutions Minds: https://minds.com/matheasysolutions Twitter: https://twitter.com/MathEasySolns Facebook: https://fb.com/MathEasySolutions LinkedIn: https://mes.fm/linkedin Pinterest: https://pinterest.com/MathEasySolns Instagram: https://instagram.com/MathEasySolutions Email me: contact@mes.fm Try our Free Calculators: https://mes.fm/calculators BMI Calculator: https://bmicalculator.mes.fm Grade Calculator: https://gradecalculator.mes.fm Mortgage Calculator: https://mortgagecalculator.mes.fm Percentage Calculator: https://percentagecalculator.mes.fm Try our Free Online Tools: https://mes.fm/tools iPhone and Android Apps: https://mes.fm/mobile-apps ... https://www.youtube.com/watch?v=68MGEV7ZLaA