{"id":362956,"date":"2023-01-07T09:18:39","date_gmt":"2023-01-07T03:48:39","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/question\/let-sbe-the-set-of-all-a-%e2%88%88r-for-which-the-angle-betwen-the-vectors-u%e2%87%80alogebi-6j3k-andv%e2%86%92logebi2j2alogebk-b1is-acute-then-s-is-equal-to\/"},"modified":"2025-06-26T14:46:02","modified_gmt":"2025-06-26T09:16:02","slug":"let-sbe-the-set-of-all-a-%e2%88%88r-for-which-the-angle-betwen-the-vectors-u%e2%87%80alogebi-6j3k-andv%e2%86%92logebi2j2alogebk-b1is-acute-then-s-is-equal-to","status":"publish","type":"questions","link":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/let-sbe-the-set-of-all-ar-for-which-the-angle-betwen-the\/","title":{"rendered":"Let \u00a0Sbe the set of all a\u00a0\u2208R for which the angle betwen the vectors\u00a0u\u21c0=alogebi^-6j^+3k^\u00a0andv\u2192=logebi^+2j^+2alogebk^\u00a0,b>1is acute. Then S is equal to"},"content":{"rendered":"","protected":false},"author":1,"template":"","meta":{"_yoast_wpseo_focuskw":"","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Let \u00a0Sbe the set of all a\u00a0\u2208R for which the angle betwen the vectors\u00a0u\u21c0=alogebi^-6j^+3k^\u00a0andv\u2192=logebi^+2j^+2alogebk^\u00a0,b>1is acute. Then S is equal to","custom_permalink":"question\/mathematics\/let-sbe-the-set-of-all-ar-for-which-the-angle-betwen-the\/"},"categories":[],"acf":{"question":"

Let  Sbe the set of all a \u2208<\/mo><\/math>R for which the angle betwen the vectors u<\/mi>\u21c0<\/mo><\/mover>=<\/mo>a<\/mi>log<\/mi>e<\/mi><\/msub>b<\/mi><\/mrow><\/mfenced>i<\/mi>^<\/mo><\/mover>-<\/mo>6<\/mn>j<\/mi>^<\/mo><\/mover>+<\/mo>3<\/mn>k<\/mi>^<\/mo><\/mover><\/math> and<\/p>

v<\/mi>\u2192<\/mo><\/mover>=<\/mo>log<\/mi>e<\/mi><\/msub>b<\/mi><\/mrow><\/mfenced>i<\/mi>^<\/mo><\/mover>+<\/mo>2<\/mn>j<\/mi>^<\/mo><\/mover>+<\/mo>2<\/mn>a<\/mi>log<\/mi>e<\/mi><\/msub>b<\/mi><\/mrow><\/mfenced>k<\/mi>^<\/mo><\/mover> <\/mo>,<\/mo>b<\/mi>><\/mo>1<\/mn><\/mrow><\/mfenced><\/math>is acute. Then S is equal to<\/p>","options":[{"option":"

\u221e<\/mo>,<\/mo> <\/mo>-<\/mo>4<\/mn>3<\/mn><\/mfrac><\/mrow><\/mfenced> <\/mspace><\/math><\/p>","correct":false},{"option":"

\u03d5<\/mi><\/math><\/p>","correct":true},{"option":"

-<\/mo>4<\/mn>3<\/mn><\/mfrac>,<\/mo> <\/mo>0<\/mn><\/mrow><\/mfenced><\/math><\/p>","correct":false},{"option":"

12<\/mn>7<\/mn><\/mfrac>,<\/mo> <\/mo>\u221e<\/mo><\/mrow><\/mfenced><\/math><\/p>","correct":false}],"solution":"

Angle  between the vectorsu<\/mi>\u2192<\/mo><\/mover>,<\/mo>v<\/mi>\u2192<\/mo><\/mover><\/math> is acute<\/p>

\u21d2<\/mo>u<\/mi>\u2192<\/mo><\/mover>.<\/mo>v<\/mi>\u2192<\/mo><\/mover>><\/mo>0<\/mn> <\/mspace>\u21d2<\/mo>a<\/mi>log<\/mi>e<\/mi><\/msub> <\/mo>b<\/mi><\/mrow><\/mfenced>2<\/mn><\/msup>-<\/mo>12<\/mn>+<\/mo>6<\/mn>a<\/mi>log<\/mi>e<\/mi><\/msub> <\/mo>b<\/mi><\/mrow><\/mfenced>><\/mo>0<\/mn> <\/mspace>\u2200<\/mo> <\/mo>b<\/mi>><\/mo>1<\/mn> <\/mspace>l<\/mi>e<\/mi>t<\/mi> <\/mo>log<\/mi>e<\/mi><\/msub>b<\/mi>=<\/mo>t<\/mi>\u21d2<\/mo>t<\/mi>><\/mo>0<\/mn> <\/mo>a<\/mi>s<\/mi> <\/mo>b<\/mi>><\/mo>1<\/mn> <\/mspace>y<\/mi>=<\/mo>a<\/mi>t<\/mi>2<\/mn><\/msup>+<\/mo>6<\/mn>a<\/mi>t<\/mi>-<\/mo>12<\/mn> <\/mo>&<\/mo> <\/mo>y<\/mi>><\/mo>0<\/mn>,<\/mo> <\/mo>\u2200<\/mo> <\/mo>t<\/mi>><\/mo>0<\/mn> <\/mspace>a<\/mi>><\/mo>0<\/mn>,<\/mo>D<\/mi><<\/mo>0<\/mn>\u21d2<\/mo>36<\/mn>a<\/mi>2<\/mn><\/msup>+<\/mo>48<\/mn>a<\/mi><<\/mo>0<\/mn> <\/mo>w<\/mi>h<\/mi>i<\/mi>c<\/mi>h<\/mi> <\/mo>i<\/mi>s<\/mi> <\/mo>n<\/mi>o<\/mi>t<\/mi> <\/mo>p<\/mi>o<\/mi>s<\/mi>s<\/mi>i<\/mi>b<\/mi>l<\/mi>e<\/mi> <\/mspace>F<\/mi>o<\/mi>r<\/mi> <\/mo>a<\/mi><<\/mo>0<\/mn>,<\/mo>D<\/mi><<\/mo>0<\/mn>\u21d2<\/mo>36<\/mn>a<\/mi>2<\/mn><\/msup>+<\/mo>48<\/mn>a<\/mi><<\/mo>0<\/mn> <\/mo>\u21d2<\/mo>12<\/mn>a<\/mi>3<\/mn>a<\/mi>+<\/mo>4<\/mn><\/mrow><\/mfenced><<\/mo>0<\/mn> <\/mspace>-<\/mo>4<\/mn><\/mrow>3<\/mn><\/mfrac><<\/mo>a<\/mi><<\/mo>0<\/mn>\u21d2<\/mo>y<\/mi><<\/mo>0<\/mn> <\/mspace>\u2234<\/mo> <\/mo>a<\/mi>\u2208<\/mo>\u03d5<\/mi><\/math><\/p>","subject":""},"yoast_head":"\nLet \u00a0Sbe the set of all a\u00a0\u2208R for which the angle betwen the vectors\u00a0u\u21c0=alogebi^-6j^+3k^\u00a0andv\u2192=logebi^+2j^+2alogebk^\u00a0,b>1is acute. Then S is equal to - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"Let \u00a0Sbe the set of all a\u00a0\u2208R for which the angle betwen the vectors\u00a0u\u21c0=alogebi^-6j^+3k^\u00a0andv\u2192=logebi^+2j^+2alogebk^\u00a0,b>1is acute. Then S is equal to\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/let-sbe-the-set-of-all-ar-for-which-the-angle-betwen-the\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Let \u00a0Sbe the set of all a\u00a0\u2208R for which the angle betwen the vectors\u00a0u\u21c0=alogebi^-6j^+3k^\u00a0andv\u2192=logebi^+2j^+2alogebk^\u00a0,b>1is acute. 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Then S is equal to - Infinity Learn by Sri Chaitanya","og_description":"Let \u00a0Sbe the set of all a\u00a0\u2208R for which the angle betwen the vectors\u00a0u\u21c0=alogebi^-6j^+3k^\u00a0andv\u2192=logebi^+2j^+2alogebk^\u00a0,b>1is acute. 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