Given the vectors A = 2i + 3j − 2k and B = −3i − j + k, determine: 2A + 3B; |4A - B|

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Hapa kuna suluhisho la swali la 1: a) Kutokana na vekta A = 2i + 3j − 2k na B = −3i − j + k. i) 2A + 3B; Step 1: Kokotoa 2A. 2A = 2(2i + 3j − 2k) = 4i + 6j − 4k Step 2: Kokotoa 3B. 3B = 3(−3i − j + k) = −9i − 3j + 3k Step 3: Jumlisha matokeo. 2A + 3B = (4i + 6j − 4k) + (−9i − 3j + 3k); 2A + 3B = (4−9)i + (6−3)j + (−4+3)k; 2A + 3B = −5i + 3j − k ii) |4A − B|; Step 1: Kokotoa 4A. 4A = 4(2i + 3j − 2k) = 8i + 12j − 8k Step 2: Kokotoa 4A − B. 4A − B = (8i + 12j − 8k) − (−3i − j + k); 4A − B = (8 − (−3))i + (12 − (−1))j + (−8 − 1)k; 4A − B = (8+3)i + (12+1)j + (−8−1)k; 4A − B = 11i + 13j − 9k Step 3: Kokotoa ukubwa wa vekta. |4A − B| = √((11)² + (13)² + (−9)²); |4A − B| = √(121 + 169 + 81); |4A − B| = √(371); |4A − B| = √(371) b) Kutokana na vekta U = 3i − 4j + 2k ni perpendicular kwa vekta V = 2i + Pj + 3k. Step 1: Tumia sheria ya dot product kwa vekta perpendicular. Ikiwa vekta mbili ni perpendicular, dot product yao ni sifuri (U · V = 0). (3)(2) + (−4)(P) + (2)(3) = 0 Step 2: Tatua kwa P. 6 − 4P + 6 = 0; 12 − 4P = 0; 4P = 12; P = 12/4; P = 3 c) Tumia uchambuzi wa vekta kubainisha eneo la pembetatu ambayo pande zake mbili ni vekta P = 2i + 2j + k na Q = i + 6j + 2k. Step 1: Kokotoa cross product ya vekta P na Q. Eneo la pembetatu iliyoundwa na vekta mbili kama pande zake ni 1/2 |P × Q|. P × Q = i & j & k 2 & 2 & 1 1 & 6 & 2 P × Q = i((2)(2) − (1)(6)) − j((2)(2) − (1)(1)) + k((2)(6) − (2)(1)); P × Q = i(4 − 6) − j(4 − 1) + k(12 − 2); P × Q = −2i − 3j + 10k Step 2: Kokotoa ukubwa wa cross product. |P × Q| = √((−2)² + (−3)² + (10)²); |P × Q| = √(4 + 9 + 100); |P × Q| = √(113) Step 3: Kokotoa eneo la pembetatu. Eneo = 1/2 |P × Q|; Eneo = 1/2√(113) units²

Given the vectors A = 2i + 3j − 2k and B = -3i - j + k, determine: 2A + 3B; |4A - B|

Answers

Step-by-step solution

Part a(i)· 3 steps

Kokotoa $2A$.

Kokotoa $3B$.

Jumlisha matokeo.

Part a(ii)· 3 steps
$\sqrt{371}$

Kokotoa $4A$.

Kokotoa $4A - B$ .

Kokotoa ukubwa wa vekta.

Part b· 2 steps3

Tumia sheria ya dot product kwa vekta perpendicular.

Tatua kwa $P$ .

Part c· 3 steps
$\frac{1}{2}\sqrt{113} \text{ units}^2$

Kokotoa cross product ya vekta $P$ na $Q$ .

Kokotoa ukubwa wa cross product.

Kokotoa eneo la pembetatu.

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