https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:cross-product-of-vectors-1/v/cross-product-of-unit-vectors 2025-05-22T13:12:03.767544849Z Cross product of unit vectors In this video, we familiarise ourselves with the formula for cross product of vectors. We look at a few scenarios and special cases. We then look at cross products of unit vectors along the axes. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Cross product of unit vectors In this video, we familiarise ourselves with the formula for cross product of vectors. We look at a few scenarios and special cases. We then look at cross products of unit vectors along the axes. https://cdn.kastatic.org/ka-youtube-converted/sjV4JvQYfSg.mp4/sjV4JvQYfSg.mp4 366 Cross product of vectors https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:cross-product-of-vectors-1/v/cross-product-in-component-form 2025-05-22T13:12:03.767544849Z Cross product in component form In this video, we learn how to take cross product of two vectors which are given in component form. We first find the cross product using the determinant formula and then understand where the determinant formula comes from. We then also figure out why the distributive property of cross product makes sense using determinants. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Cross product in component form In this video, we learn how to take cross product of two vectors which are given in component form. We first find the cross product using the determinant formula and then understand where the determinant formula comes from. We then also figure out why the distributive property of cross product makes sense using determinants. https://cdn.kastatic.org/ka-youtube-converted/FMdC9nFf-N0.mp4/FMdC9nFf-N0.mp4 315 Cross product of vectors https://www.khanacademy.org/math/grade-11-math-snc-aligned/x07cdd52586d25c43:vectors-in-space/x07cdd52586d25c43:scalar-triple-product-of-vectors-definition-component-form-and-properties/e/cross-product-of-unit-vectors 2025-08-08T02:45:51.78671331Z Cross product of unit vectors Cross product of unit vectors Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:cross-product-of-vectors-1/v/finding-perpendicular-vectors-using-cross-product 2025-05-22T13:12:03.767544849Z Finding perpendicular vectors using cross product In this video we try to find vectors perpendicular to a pair of given vectors. Since cross products give exactly this, we take the cross product of given vectors. Additionally, if we want to find the unit vectors, we divide the cross product by its magnitude. We also solve a problem where the dot product of the perpendicular is given and we have to find the missing vector. Note - in general, there are two vectors perpendicular to two given vectors - both opposite of each other. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Finding perpendicular vectors using cross product In this video we try to find vectors perpendicular to a pair of given vectors. Since cross products give exactly this, we take the cross product of given vectors. Additionally, if we want to find the unit vectors, we divide the cross product by its magnitude. We also solve a problem where the dot product of the perpendicular is given and we have to find the missing vector. Note - in general, there are two vectors perpendicular to two given vectors - both opposite of each other. https://cdn.kastatic.org/ka-youtube-converted/PPkjpE2p7QY.mp4/PPkjpE2p7QY.mp4 500 Cross product of vectors https://www.khanacademy.org/math/grade-11-math-snc-aligned/x07cdd52586d25c43:vectors-in-space/x07cdd52586d25c43:scalar-triple-product-of-vectors-definition-component-form-and-properties/e/finding-perpendicular-vectors-using-cross-product 2025-08-08T02:45:51.78671331Z Finding perpendicular vectors using cross product Finding perpendicular vectors using cross product Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:cross-product-of-vectors-1/v/finding-area-using-cross-product 2025-05-22T13:12:03.767544849Z Finding area using cross product In this video, we derive the formula for the area of a parallelogram using the cross product. We then solve problems where we use the formula to find the area of a triangle, a parallelogram, and a rectangle. In all these problems (except the last one), we first find the vectors representing the adjacent sides, then find their cross product, and then take its magnitude to get to the area. For the last one, we leverage our visualisation skills. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Finding area using cross product In this video, we derive the formula for the area of a parallelogram using the cross product. We then solve problems where we use the formula to find the area of a triangle, a parallelogram, and a rectangle. In all these problems (except the last one), we first find the vectors representing the adjacent sides, then find their cross product, and then take its magnitude to get to the area. For the last one, we leverage our visualisation skills. https://cdn.kastatic.org/ka-youtube-converted/WMwzBbhhLcU.mp4/WMwzBbhhLcU.mp4 420 Cross product of vectors https://www.khanacademy.org/math/grade-11-math-snc-aligned/x07cdd52586d25c43:vectors-in-space/x07cdd52586d25c43:scalar-triple-product-of-vectors-definition-component-form-and-properties/e/finding-area-using-cross-product 2025-08-08T02:45:51.78671331Z Finding area using cross product Finding area using cross product Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:cross-product-of-vectors-1/v/conceptual-questions-on-product-of-vectors 2025-05-22T13:12:03.767544849Z Conceptual questions on product of vectors In this video, we test our fundamentals on dot and cross product by answering some conceptual questions. Some examples of the questions discussed in this video include: If one vector is zero, what can we say about the other vector? Are vectors zero when dot product is zero? Are vectors zero when cross product is zero? When are both the products equal? Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Conceptual questions on product of vectors In this video, we test our fundamentals on dot and cross product by answering some conceptual questions. Some examples of the questions discussed in this video include: If one vector is zero, what can we say about the other vector? Are vectors zero when dot product is zero? Are vectors zero when cross product is zero? When are both the products equal? https://cdn.kastatic.org/ka-youtube-converted/aU3FQWHs8qE.mp4/aU3FQWHs8qE.mp4 572 Cross product of vectors https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:cross-product-of-vectors-1/e/conceptual-questions-on-product-of-vectors 2025-08-08T02:45:51.78671331Z Conceptual questions on product of vectors Conceptual questions on product of vectors Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:cross-product-of-vectors-1/v/manipulating-equations-involving-products-of-vectors 2025-05-22T13:12:03.767544849Z Manipulating equations involving products of vectors In this video, we solve problems that apply dot product of vectors. In the first problem, we're given information about the magnitudes of the vectors and their perpendicularity. We're asked to find the magnitude of sum of these vectors. In the second problem, we're given the sum and we're trying to find the sum of pairwise products. In the final problem, we're given two vectors and we're asked to break the second one into two parts - one perpendicular and the other parallel to the first one. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Manipulating equations involving products of vectors In this video, we solve problems that apply dot product of vectors. In the first problem, we're given information about the magnitudes of the vectors and their perpendicularity. We're asked to find the magnitude of sum of these vectors. In the second problem, we're given the sum and we're trying to find the sum of pairwise products. In the final problem, we're given two vectors and we're asked to break the second one into two parts - one perpendicular and the other parallel to the first one. https://cdn.kastatic.org/ka-youtube-converted/0rQvihZq5O8.mp4/0rQvihZq5O8.mp4 490 Cross product of vectors https://www.khanacademy.org/math/grade-11-math-snc-aligned/x07cdd52586d25c43:vectors-in-space/x07cdd52586d25c43:solve-situations-in-daily-life-based-on-cross-or-dot-vector-product/e/manipulating-equations-involving-products-of-vectors 2025-08-08T02:45:51.78671331Z Manipulating equations involving products of vectors Manipulating equations involving products of vectors Ashish Gupta exercise https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/linear-algebra-cross-product-introduction 2020-07-26T09:23:49Z Cross product introduction Introduction to the cross product Sal Khan video https://cdn.kastatic.org/googleusercontent/P6fTzoK6rLVVQF4eua0JeaGV21oApezgfDFJUpaeJfx83aMRo1KivA_pvxHzGK1A0sYWLOWgn7aKLlsn5PGJC7o Cross product introduction Introduction to the cross product https://cdn.kastatic.org/ka-youtube-converted/pJzmiywagfY.mp4/pJzmiywagfY.mp4 947 Cross product of vectors https://www.khanacademy.org/science/in-in-class11th-physics/in-in-system-of-particles-and-rotational-motion/in-in-cross-product/v/cross-product-1 2021-05-17T03:42:37Z Cross product 1 Introduction to the cross product Sal Khan video https://cdn.kastatic.org/googleusercontent/1dyRhyEsJB4tyLdePnXPP6yCsoQ3qzcPxXrkoZp2Tn2x3S65U3REboU7jPUy2QULW-0gMk4ods7hGKkSORtAA_7v0A Cross product 1 Introduction to the cross product https://cdn.kastatic.org/ka-youtube-converted/zA0fvwtvgvA.mp4/zA0fvwtvgvA.mp4 602 Cross product of vectors https://www.khanacademy.org/science/in-in-class11th-physics/in-in-system-of-particles-and-rotational-motion/in-in-cross-product/v/cross-product-2 2021-05-17T03:42:39Z Cross product 2 A little more intuition on the cross product. Sal Khan video https://cdn.kastatic.org/googleusercontent/hterZg1rl5CpYA0mA-xkMGL9ie1tu0KODpCFM5Zw-ZsANODpxE9cDY_SmfqKLd87aMoUQPyZwt8OsSOkyJHyKShx Cross product 2 A little more intuition on the cross product. https://cdn.kastatic.org/ka-youtube-converted/o_puKe_lTKk.mp4/o_puKe_lTKk.mp4 653 Cross product of vectors https://www.khanacademy.org/math/grade-11-math-snc-aligned/x07cdd52586d25c43:vectors-in-space/x07cdd52586d25c43:applications-of-cross-product/e/cross-product-of-two-vectors 2026-04-07T23:36:17.218592739Z Cross product of two vectors cross product exercise https://www.khanacademy.org/math/grade-11-math-snc-aligned/x07cdd52586d25c43:vectors-in-space/x07cdd52586d25c43:applications-of-cross-product/e/cross-product-of-two-vectors-advanced 2026-04-07T22:59:30.107127435Z Cross product of two vectors (advanced) cross product - find perpendicular vector, properties of vector product and area of parallelogram exercise