https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:untitled-1286/v/dot-product-of-vectors-in-3d 2025-05-22T13:12:03.767544849Z Dot product of vectors in 3D In this video, we familiarise ourselves with the formula for dot product of vectors. We look at a few scenarios and special cases. We then look at dot products of unit vectors along the axes. We look at how do we get the dot product of two vectors given in component form. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Dot product of vectors in 3D In this video, we familiarise ourselves with the formula for dot product of vectors. We look at a few scenarios and special cases. We then look at dot products of unit vectors along the axes. We look at how do we get the dot product of two vectors given in component form. https://cdn.kastatic.org/ka-youtube-converted/XK8iPlmKpWA.mp4/XK8iPlmKpWA.mp4 588 Dot product of vectors https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:untitled-1286/v/angle-between-vectors 2025-05-22T13:12:03.767544849Z Angle between vectors We look at ways to get information about the angle between two vectors from their dot product. We find the angle between vectors given in their component form. We then solve problems where magnitude of vectors are given and we need to know the angle between them. Finally, we look at a challenging problem where we apply this skill to find the angle between two unit vectors. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Angle between vectors We look at ways to get information about the angle between two vectors from their dot product. We find the angle between vectors given in their component form. We then solve problems where magnitude of vectors are given and we need to know the angle between them. Finally, we look at a challenging problem where we apply this skill to find the angle between two unit vectors. https://cdn.kastatic.org/ka-youtube-converted/hR3vpW__HTM.mp4/hR3vpW__HTM.mp4 472 Dot product of vectors https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:untitled-1286/e/angle-between-two-vectors 2025-08-08T02:45:51.78671331Z Angle between two vectors Angle between two vectors Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:untitled-1286/v/finding-angles-of-a-triangle 2025-05-22T13:12:03.767544849Z Finding angles of a triangle In this video, we learn the skill of finding the angles of a triangle using dot products. We first find the vectors along the sides, then take their dot products. We also find the magnitudes of these vectors. Once the raw materials are ready, we divide the dot products with product of magnitudes to find cos of angles. This video also dives deep into the meaning behind the negative signs of cos products. We finally look at a cleaner way to pick vectors along the sides in such scenarios. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Finding angles of a triangle In this video, we learn the skill of finding the angles of a triangle using dot products. We first find the vectors along the sides, then take their dot products. We also find the magnitudes of these vectors. Once the raw materials are ready, we divide the dot products with product of magnitudes to find cos of angles. This video also dives deep into the meaning behind the negative signs of cos products. We finally look at a cleaner way to pick vectors along the sides in such scenarios. https://cdn.kastatic.org/ka-youtube-converted/dMVOhEOZ5SM.mp4/dMVOhEOZ5SM.mp4 652 Dot product of vectors https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:untitled-1286/e/finding-angles-of-a-triangle 2025-08-08T02:45:51.78671331Z Finding angles of a triangle Finding angles of a triangle Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:untitled-1286/v/solving-vectors-equations-involving-dot-products 2025-05-22T13:12:03.767544849Z Solving vectors equations involving dot products In this video, we solve vector equations involving dot products. We start with finding the magnitude of a missing vector. We take the dot product and solve for the unknown. We then solve a problem where we get a linear equation in the unknown variable. Finally, we solve a multi step problem where add vectors, take magnitude, find the unit vector, and then solve for the unknown. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Solving vectors equations involving dot products In this video, we solve vector equations involving dot products. We start with finding the magnitude of a missing vector. We take the dot product and solve for the unknown. We then solve a problem where we get a linear equation in the unknown variable. Finally, we solve a multi step problem where add vectors, take magnitude, find the unit vector, and then solve for the unknown. https://cdn.kastatic.org/ka-youtube-converted/hwO0mo5yQGU.mp4/hwO0mo5yQGU.mp4 461 Dot product of vectors https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:untitled-1286/e/solving-vectors-equations-involving-dot-products 2025-08-08T02:45:51.78671331Z Solving vectors equations involving dot products Solving vectors equations involving dot products Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:untitled-1286/v/cauchy-schwartz-and-triangle-inequality-of-vectors 2025-05-22T13:12:03.767544849Z Cauchy-Schwartz and Triangle inequality of vectors In this video, we discuss inequalities of multiplication and addition of vectors. Cauchy-Schwartz inequality says that the magnitude of the dot product can never be greater than the product of magnitudes. Triangle inequality says that the magnitude of the sum of vectors can never be greater than the sum of magnitudes of vectors. We also discuss the edge cases - when do both sides become equal. We then discuss their use cases - which one to use if we're proving collinearity and which one to use if we're proving that vectors are parallel. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Cauchy-Schwartz and Triangle inequality of vectors In this video, we discuss inequalities of multiplication and addition of vectors. Cauchy-Schwartz inequality says that the magnitude of the dot product can never be greater than the product of magnitudes. Triangle inequality says that the magnitude of the sum of vectors can never be greater than the sum of magnitudes of vectors. We also discuss the edge cases - when do both sides become equal. We then discuss their use cases - which one to use if we're proving collinearity and which one to use if we're proving that vectors are parallel. https://cdn.kastatic.org/ka-youtube-converted/Ct9AAE_aM2o.mp4/Ct9AAE_aM2o.mp4 630 Dot product of vectors https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:untitled-1286/e/cauchy-schwartz-and-triangle-inequality-of-vectors 2025-08-08T02:45:51.78671331Z Cauchy-Schwartz and Triangle inequality of vectors Cauchy-Schwartz and Triangle inequality of vectors Ashish Gupta exercise https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:untitled-1286/v/projection-of-a-vector 2025-05-22T13:12:03.767544849Z Projection of a vector In this video, we discuss the concept of projection. We try to find the projection of a given vector on another. We then apply what we observe to derive the formula for projection and projection vector. We finally apply the formula on a practice problem. Ashish Gupta video https://cdn.kastatic.org/googleusercontent/ZCdwTudJg6e6n-P2gsaUborP4izvMsGo71pvEVlX9dNYWcLXcP7VHkWpn2grt4TUP1KoJLQP9NswyHBuBLSFTBw Projection of a vector In this video, we discuss the concept of projection. We try to find the projection of a given vector on another. We then apply what we observe to derive the formula for projection and projection vector. We finally apply the formula on a practice problem. https://cdn.kastatic.org/ka-youtube-converted/w9ED_UMIWO4.mp4/w9ED_UMIWO4.mp4 518 Dot product of vectors https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/vector-dot-product-and-vector-length 2021-03-09T15:59:33Z Vector dot product and vector length Definitions of the vector dot product and vector length Sal Khan video https://cdn.kastatic.org/googleusercontent/5PsMopHA-5Av4_payq7p7AkmNzVa_u5ZM_D03dCVsjMbzhVLK99132nP0S276nkeaH8oFIPlS4Z_1DSaT_01x9w Vector dot product and vector length Definitions of the vector dot product and vector length https://cdn.kastatic.org/ka-youtube-converted/WNuIhXo39_k.mp4/WNuIhXo39_k.mp4 550 Dot product of vectors https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:untitled-1286/e/dot-product-of-two-vectors 2025-05-05T08:14:25.350149109Z Dot product of two vectors dot product exercise https://www.khanacademy.org/math/grade-11-math-snc-aligned/x07cdd52586d25c43:vectors-in-space/x07cdd52586d25c43:applications-of-the-dot-product/e/projection-of-a-vector-on-a-line 2026-04-08T00:54:09.631945657Z Projection of a vector on a line projection of vectors exercise https://www.khanacademy.org/math/ka-math-class-12/x6f66cbee9a2f805b:vector-algebra-ncert-new/x6f66cbee9a2f805b:untitled-1286/e/dot-product-of-two-vectors-advanced 2026-04-08T00:36:09.500768563Z Dot product of two vectors (advanced) dp exercise