[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-523550-105":59,"doc-detail-523550-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","three-dimensional-vector-cross-products-vector-cross-products-worksheet","Three-Dimensional Vector Cross Products - Vector Cross Products Worksheet","","Three-dimensional vector worksheets focused on the cross product and its applications in geometry. Problems ask students to compute a × b, AB × CD, and to find a perpendicular vector from two given vectors. Additional tasks determine triangle areas using adjacent side vectors and given vertices, then find parallelogram area from adjacent side vectors. The set concludes with parallelepiped volume using adjacent edges, reinforced through repeated coordinate-based vector data.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/three-dimensional-vector-cross-products-vector-cross-products-worksheet/523550/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/three-dimensional-vector-cross-products-vector-cross-products-worksheet/523550.png","ImageObject",300,407,{"name":92,"@type":93},"ahduag","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-11","2026-10-08",true,{"@type":102,"interactionType":103,"userInteractionCount":29},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What does the worksheet ask you to do with the cross product?","Question",{"text":112,"@type":113},"It asks you to compute cross products such as a × b and AB × CD, and to use them to find perpendicular relationships between vectors.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"How are triangle areas determined in the worksheet?",{"text":117,"@type":113},"Triangle areas are found using the given vectors as two adjacent sides, and in another set using the given vertices’ coordinates.",{"name":119,"@type":110,"acceptedAnswer":120},"How is parallelepiped volume calculated here?",{"text":121,"@type":113},"Volume is determined using the given vectors as adjacent edges, using the coordinate data provided for each vector.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},523550,1791774917,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":29,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":143,"read_time":52,"preview_page_count":4},3985747870414,"https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc","Kuta Software-Infinite Precalculus  \nThree-Dimensional Vector Cross Products  \nName  Date  Period   \nFind the cross product of the given vectors.  \n1) a × b  \na 􀀠 􀀐􀀜 , 􀀘 , 􀀐􀀖 b 􀀠 􀀕 , 􀀜 , 􀀕  \n2) AB × CD  \nGiven: A = (􀀐􀀙 , 􀀐􀀕 , 􀀙) B = (􀀐􀀘 , 􀀐􀀘 , 􀀐􀀖)  \nC = (􀀛 , 􀀚 , 􀀐􀀙) D = (􀀐􀀘 , 􀀐􀀗 , 􀀐􀀔)  \nFind a vector that is perpendicular to the given vectors.  \n3) u 􀀠 􀀐􀀘 , 􀀙 , 􀀚 4) RS and RT  \nv   􀀠 􀀔 , 􀀐􀀜 , 􀀓 Given: R = (􀀖 , 􀀚 , 􀀕) S = (􀀔 , 􀀚 , 􀀐􀀚)  \nT = (􀀐􀀔 , 􀀐􀀘 , 􀀙)  \nFind the area of a triangle with the given vectors as two adjacent sides.  \n5) a 􀀠 􀀜 , 􀀐􀀕 , 􀀜 b 􀀠 􀀗 , 􀀜 , 􀀐􀀕  \n6) u 􀀠 􀀓 , 􀀕 , 􀀐􀀔  v 􀀠 􀀐􀀖 , 􀀙 , 􀀛  \nFind the area of a triangle with the given vertices.  \n7) R = (􀀖 , 􀀐􀀕 , 􀀚)  \nS = (􀀓 , 􀀐􀀚 , 􀀜)  \nT = (􀀙 , 􀀐􀀚 , 􀀜)  \n8) A = (􀀙 , 􀀜 , 􀀐􀀛)  \nB = (􀀓 , 􀀐􀀜 , 􀀐􀀔)  \nC = (􀀐􀀛 , 􀀐􀀜 , 􀀕)  \nFind the area of a parallelogram with the given vectors as two adjacent sides.  \n9) u   􀀠 􀀐􀀚 , 􀀔 , 􀀓 10) AB and AC  \nv 􀀠 􀀚 , 􀀐􀀘 , 􀀐􀀔  Given: A = (􀀐􀀛 , 􀀘 , 􀀐􀀛) B = (􀀗 , 􀀛 , 􀀐􀀛)  \nC = (􀀙 , 􀀐􀀖 , 􀀛)  \nFind the volume of a parallelepiped with the given vectors as adjacent edges.  \n11) u   􀀠 􀀐􀀙 , 􀀗 , 􀀐􀀜 12) TX , TY , and TZ  \nv 􀀠 􀀕 , 􀀐􀀘 , 􀀛 Given: T = (􀀓 , 􀀐􀀗 , 􀀐􀀛) X = (􀀛 , 􀀔 , 􀀐􀀜)  \nw   􀀠 􀀓 , 􀀐􀀗 , 􀀛 Y = (􀀐􀀔 , 􀀐􀀘 , 􀀐􀀕) Z = (􀀓 , 􀀐􀀛 , 􀀖)  \nKuta Software-Infinite Precalculus  \nThree-Dimensional Vector Cross Products  \nName  Date  Period   \nFind the cross product of the given vectors.  \n1) a × b  \na 􀀠 􀀐􀀜 , 􀀘 , 􀀐􀀖 b 􀀠 􀀕 , 􀀜 , 􀀕  \n􀀖􀀚 , 􀀔􀀕 , 􀀐􀀜􀀔   \n2) AB × CD  \nGiven: A = (􀀐􀀙 , 􀀐􀀕 , 􀀙) B = (􀀐􀀘 , 􀀐􀀘 , 􀀐􀀖)  \nC = (􀀛 , 􀀚 , 􀀐􀀙) D = (􀀐􀀘 , 􀀐􀀗 , 􀀐􀀔)  \n􀀐􀀔􀀔􀀗 , 􀀔􀀔􀀕 , 􀀐􀀘􀀓  \nFind a vector that is perpendicular to the given vectors.  \n3) u 􀀠 􀀐􀀘 , 􀀙 , 􀀚 v 􀀠 􀀔 , 􀀐􀀜 , 􀀓  \n􀀐􀀙􀀖 , 􀀐􀀚 , 􀀐􀀖􀀜  \n4) RS and RT  \nGiven: R = (􀀖 , 􀀚 , 􀀕) S = (􀀔 , 􀀚 , 􀀐􀀚)  \nT = (􀀐􀀔 , 􀀐􀀘 , 􀀙)  \n􀀔􀀓􀀛 , 􀀐􀀗􀀗 , 􀀐􀀕􀀗  \nFind the area of a triangle with the given vectors as two adjacent sides.  \n5) a 􀀠 􀀜 , 􀀐􀀕 , 􀀜 b 􀀠 􀀗 , 􀀜 , 􀀐􀀕  \n6) u 􀀠 􀀓 , 􀀕 , 􀀐􀀔  v 􀀠 􀀐􀀖 , 􀀙 , 􀀛  \n􀀔􀀙􀀚􀀙􀀙  \n􀀕  \n􀁼 􀀙􀀗􀀑􀀚􀀗􀀕  \n􀀕 units  \n 􀀕􀀖   \n􀀕  \n􀀠 􀀔􀀔􀀑􀀘  \n􀀕 units  \nFind the area of a triangle with the given vertices.  \n7) R = (􀀖 , 􀀐􀀕 , 􀀚) S = (􀀓 , 􀀐􀀚 , 􀀜) T = (􀀙 , 􀀐􀀚 , 􀀜)  \n􀀖 􀀕􀀜 􀁼 􀀔􀀙􀀑􀀔􀀘􀀘 units 􀀕  \n8) A = (􀀙 , 􀀜 , 􀀐􀀛)  \nB = (􀀓 , 􀀐􀀜 , 􀀐􀀔)  \nC = (􀀐􀀛 , 􀀐􀀜 , 􀀕)  \n􀀙􀀕􀀚􀀗 􀁼 􀀚􀀜􀀑􀀕􀀓􀀜 units 􀀕  \nFind the area of a parallelogram with the given vectors as two adjacent sides.  \n9) u 􀀠 􀀐􀀚 , 􀀔 , 􀀓 v 􀀠 􀀚 , 􀀐􀀘 , 􀀐􀀔   \n􀀛􀀖􀀗 􀁼 􀀕􀀛􀀑􀀛􀀚􀀜  \n􀀕 units  \n10) AB and AC  \nGiven: A = (􀀐􀀛 , 􀀘 , 􀀐􀀛) B = (􀀗 , 􀀛 , 􀀐􀀛)  \nC = (􀀙 , 􀀐􀀖 , 􀀛)  \n􀀗􀀕 􀀖􀀖 􀁼 􀀕􀀗􀀔􀀑􀀕􀀚􀀕 units 􀀕  \nFind the volume of a parallelepiped with the given vectors as adjacent edges.  \n11) u   􀀠 􀀐􀀙 , 􀀗 , 􀀐􀀜 12) TX , TY , and TZ  \nv 􀀠 􀀕 , 􀀐􀀘 , 􀀛 Given: T = (􀀓 , 􀀐􀀗 , 􀀐􀀛) X = (􀀛 , 􀀔 , 􀀐􀀜)  \nw   􀀠 􀀓 , 􀀐􀀗 , 􀀛 Y = (􀀐􀀔 , 􀀐􀀘 , 􀀐􀀕) Z = (􀀓 , 􀀐􀀛 , 􀀖)  \n􀀘􀀙 units 􀀖 􀀔􀀘􀀘 units 􀀖  \nCreate your own worksheets like this one with Infinite Precalculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCaikIjYfleM8SO","https://ap.wps.com/l/cbCaikIjYfleM8SO","pdf",33849,"English","# Find the cross product of vectors\n## Compute a × b and AB × CD\n## Find a perpendicular vector\n# Find areas using vector geometry\n## Triangle area from adjacent side vectors\n## Triangle area from given vertices\n## Parallelogram area from adjacent side vectors\n# Find volume from adjacent edge vectors","[{\"question\":\"What does the worksheet ask you to do with the cross product?\",\"answer\":\"It asks you to compute cross products such as a × b and AB × CD, and to use them to find perpendicular relationships between vectors.\"},{\"question\":\"How are triangle areas determined in the worksheet?\",\"answer\":\"Triangle areas are found using the given vectors as two adjacent sides, and in another set using the given vertices’ coordinates.\"},{\"question\":\"How is parallelepiped volume calculated here?\",\"answer\":\"Volume is determined using the given vectors as adjacent edges, using the coordinate data provided for each vector.\"}]","Three-Dimensional Vector Cross Products - Vector Cross Products Worksheet | PDF",1791426996]